You can run the test with the following command:

$ ./out/Default/browser_tests --gtest_filter="TabIndex/FilesAppBrowserTest.Test/tabindexSaveFileDialogDownloads"     --enable-features=WebUIPolymer2  --enable-pixel-output-in-tests --use-gpu-in-tests --ui-test-action-timeout=1000000 --enable-file-manager-step-by-step-tests --webui-test-wait-debugger --remote-debugging-port=9222

See attached the screenshot of a failing test.

The log of a failing test:

IMPORTANT DEBUGGING NOTE: each test is run inside its own process.                                                                                                                                                                                                                        
For debugging a test inside a debugger, use the                                                                                                                                                                                                                                           
--gtest_filter=<your_test_name> flag along with either                                                                                                                                                                                                                                    
--single_process (to run the test in one launcher/browser process) or                                                                                                                                                                                                                     
--single-process (to do the above, and also run Chrome in single-process mode).                                                                                                                                                                                                           
Using sharding settings from environment. This is shard 0/1                                                                                                                                                                                                                               
Using 1 parallel jobs.                                                                                                                                                                                                                                                                    
Note: Google Test filter = TabIndex/FilesAppBrowserTest.Test/tabindexSaveFileDialogDownloads                                                                                                                                                                                              
[==========] Running 1 test from 1 test case.                                                                                                                                                                                                                                             
[----------] Global test environment set-up.                                                                                                                                                                                                                                              
[----------] 1 test from TabIndex/FilesAppBrowserTest                                                                                                                                                                                                                                     
[ RUN      ] TabIndex/FilesAppBrowserTest.Test/tabindexSaveFileDialogDownloads                                                                                                                                                                                                            
                                                                                                                                                                                                                                                                                          
DevTools listening on ws://127.0.0.1:9222/devtools/browser/98718bde-10f3-416c-a5ed-6ab2b315171f                                                                                                                                                                                           
[38831:38831:0913/185312.236547:ERROR:gpu_interface_provider.cc(85)] Not implemented reached in virtual void content::GpuInterfaceProvider::RegisterOzoneGpuInterfaces(service_manager::BinderRegistry *)
[38831:38831:0913/185312.903813:INFO:file_manager_browsertest_base.cc(1118)] FileManagerBrowserTest::StartTest tabindexSaveFileDialogDownloads
[38831:38831:0913/185313.283840:INFO:CONSOLE(5675)] "Requesting volume list.", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (5675)
[38831:38831:0913/185313.283973:INFO:CONSOLE(1159)] "Cache database creating or upgrading.", source: chrome-extension://pmfjbimdmchhbnneeidfognadeopoehp/background_scripts.js (1159)
[38831:38831:0913/185313.285042:INFO:CONSOLE(5677)] "Volume list fetched with: 3 items.", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (5677)
[38831:38831:0913/185313.285064:INFO:CONSOLE(5687)] "Initializing volume: android_files:AndroidFiles", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (5687)
[38831:38831:0913/185313.285432:INFO:CONSOLE(6081)] "Requesting file system.", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (6081)
[38831:38831:0913/185313.286820:INFO:CONSOLE(5687)] "Initializing volume: downloads:Downloads", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (5687)
[38831:38831:0913/185313.286914:INFO:CONSOLE(6081)] "Requesting file system.", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (6081)
[38831:38831:0913/185313.287014:INFO:CONSOLE(5687)] "Initializing volume: drive:drive-user", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (5687)
[38831:38831:0913/185313.287088:INFO:CONSOLE(6081)] "Requesting file system.", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (6081)
[38831:38831:0913/185313.288758:INFO:CONSOLE(6129)] "File system obtained: android_files:AndroidFiles", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (6129)
[38831:38831:0913/185313.290100:INFO:CONSOLE(5691)] "Initialized volume: android_files:AndroidFiles", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (5691)
[38831:38831:0913/185313.290286:INFO:CONSOLE(6129)] "File system obtained: downloads:Downloads", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (6129)
[38831:38831:0913/185313.290765:INFO:CONSOLE(5691)] "Initialized volume: downloads:Downloads", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (5691)
[38831:38831:0913/185313.290920:INFO:CONSOLE(6129)] "File system obtained: drive:drive-user", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (6129)
[38831:38831:0913/185313.291591:INFO:CONSOLE(5691)] "Initialized volume: drive:drive-user", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (5691)
[38831:38831:0913/185313.291612:INFO:CONSOLE(5695)] "Initialized all volumes.", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (5695)
[38831:38831:0913/185313.384032:INFO:CONSOLE(0)] "HTML Imports is deprecated and will be removed in M73, around March 2019. Please use ES modules instead. See https://www.chromestatus.com/features/5144752345317376 for more details.", source:  (0)
[38831:38831:0913/185313.585114:INFO:CONSOLE(1283)] "document.registerElement is deprecated and will be removed in M73, around March 2019. Please use window.customElements.define instead. See https://www.chromestatus.com/features/4642138092470272 for more details.", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/foreground/js/main_scripts.js (1283)
[38831:38831:0913/185313.930602:INFO:CONSOLE(12433)] "Unknown error.", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/foreground/js/main_scripts.js (12433)
[38831:38831:0913/185313.951992:INFO:CONSOLE(18700)] "Uncaught (in promise) Error retrieving Web Store access token.", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/foreground/js/main_scripts.js (18700)
[38831:38831:0913/185314.175325:INFO:CONSOLE(4960)] "Waiting for the result of selectVolume", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (4960)
[38831:38831:0913/185314.189159:INFO:CONSOLE(4957)] "Received the result of selectVolume", source: chrome-extension://hhaomjibdihmijegdhdafkllkbggdgoj/background/js/background_common_scripts.js (4957)
[38831:38831:0913/185317.255385:INFO:CONSOLE(156)] "at RemoteCallFilesApp.RemoteCall.waitForElement (/remote_call.js:182:15): Element #filename-input-textbox:focus-within is not found.", source: chrome-extension://oobinhbdbiehknkpbpejbbpdbkdjmoco/test_util.js (156)
[38831:38831:0913/185320.296397:INFO:CONSOLE(156)] "at RemoteCallFilesApp.RemoteCall.waitForElement (/remote_call.js:182:15): Element #filename-input-textbox:focus-within is not found.", source: chrome-extension://oobinhbdbiehknkpbpejbbpdbkdjmoco/test_util.js (156)
[38831:38831:0913/185323.338445:INFO:CONSOLE(156)] "at RemoteCallFilesApp.RemoteCall.waitForElement (/remote_call.js:182:15): Element #filename-input-textbox:focus-within is not found.", source: chrome-extension://oobinhbdbiehknkpbpejbbpdbkdjmoco/test_util.js (156)
Still waiting for the following processes to finish:
        ./out/cOS/browser_tests --disable-gpu-process-for-dx12-vulkan-info-collection --enable-features=WebUIPolymer2 --enable-pixel-output-in-tests --gtest_also_run_disabled_tests --gtest_filter=TabIndex/FilesAppBrowserTest.Test/tabindexSaveFileDialogDownloads --remote-debugging-port=9222 --single_process --test-launcher-output=/tmp/.org.chromium.Chromium.n8NPVQ/resultstNbvND/test_results.xml --ui-test-action-timeout=1000000 --use-gpu-in-tests --user-data-dir=/tmp/.org.chromium.Chromium.n8NPVQ/dpxIEmK
[38831:38831:0913/185326.382701:INFO:CONSOLE(156)] "at RemoteCallFilesApp.RemoteCall.waitForElement (/remote_call.js:182:15): Element #filename-input-textbox:focus-within is not found.", source: chrome-extension://oobinhbdbiehknkpbpejbbpdbkdjmoco/test_util.js (156)
...

Deleted: Selection_052.png 68.1 KB

	Selection_052.png 68.1 KB View Download

Briefly looked at the issue. A couple of weird issues stand out in the DevTools CSS inspector.

See css_malformed.png, where the CSS for the cr-input is wrong
1) multiple variable declarations appear in one line
2) cr-input:unresolved is applied, which should only apply if cr-input.html has not been loaded yet.

See css_correct_p1.png for how the CSS should look like.

Deleted: css_malformed.png 36.2 KB

	css_malformed.png 36.2 KB View Download

Deleted: css_correct_p1.png 32.9 KB

	css_correct_p1.png 32.9 KB View Download

FYI, I was able to locally fix the malformed CSS issue with https://chromium-review.googlesource.com/c/chromium/src/+/1244833.

<style is="custom-style"> is not supported in Polymer 2, and needs to be tweaked (here and elsewhere where it occurs). This is covered at [1] (search for "Wrap custom-style elements
".

The problem with the unexpected presence of :unresolved still happens though, see attachment.

@noel: Any ideas?

[1] https://www.polymer-project.org/2.0/docs/upgrade#shadow-dom-styles

Deleted: css_malformed_but_better.png 24.3 KB

	css_malformed_but_better.png 24.3 KB View Download

Found the answer, at [1]. Basically :unresolved does not work with Shadow DOM v1 anymore, and :not(:defined) is the new way of preventing FOUC.

[1] https://github.com/Polymer/docs/issues/2297#issuecomment-326465723

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/247bda68792ed390290271138de6c334f58d33ed

commit 247bda68792ed390290271138de6c334f58d33ed
Author: dpapad <dpapad@chromium.org>
Date: Thu Sep 27 00:59:13 2018

Files App: Fix occurence of <style is="custom-style"> to work with Polymer 2.

 - wrap <style is="custom-style"> with <custom-style>
 - move custom-style inside <body>

Bug:  884030 
Change-Id: Ic45df6d1846fdf6f6ee0cc5e23523779489bdb34
Reviewed-on: https://chromium-review.googlesource.com/1244833
Commit-Queue: Luciano Pacheco <lucmult@chromium.org>
Reviewed-by: Luciano Pacheco <lucmult@chromium.org>
Cr-Commit-Position: refs/heads/master@{#594548}
[modify] https://crrev.com/247bda68792ed390290271138de6c334f58d33ed/ui/file_manager/file_manager/main.html

Now that  issue 889336  has been fixed, the cr-input appears as expected. Marking as fixed. Will re-enable the tests as part of the parent bug.