Steps to reproduce the problem:
1. Build ChromiumPublic.apk
2. Turn off wifi on the device
3. Run the test:
out-gn/chromium-android/bin/run_chrome_public_test_apk --gtest-filter=ClearBrowsingDataPreferencesTest#testImportantSitesDialogNoFiltering

What is the expected behavior?
Test to pass.

What went wrong?
Test failed with error:
chromium: [INFO:CONSOLE(0)] "Blocked script execution in 'http://127.0.0.1:39719/chrome/test/data/android/storage_persistance.html' because the document's frame is sandboxed and the 'allow-scripts' permission is not set.", source: http://127.0.0.1:39719/chrome/test/data/android/storage_persistance.html (0)

Did this work before? No 

Chrome version: r537781  Channel: dev
OS Version: Nougat mr2
Flash Version: 

It happens that test testImportantSitesDialogNoFiltering fails when wifi is turned off if the page is completely loaded before important dialog is shown.

The test fails with error: 
chromium: [INFO:CONSOLE(0)] "Blocked script execution in 'http://127.0.0.1:39719/chrome/test/data/android/storage_persistance.html' because the document's frame is sandboxed and the 'allow-scripts' permission is not set.", source: http://127.0.0.1:39719/chrome/test/data/android/storage_persistance.html (0)

This actually happens due to page being completely loaded in loadUrl():
https://cs.chromium.org/chromium/src/chrome/android/javatests/src/org/chromium/chrome/browser/preferences/privacy/ClearBrowsingDataPreferencesTest.java?l=418

before important dialog is being shown:
https://cs.chromium.org/chromium/src/chrome/android/javatests/src/org/chromium/chrome/browser/preferences/privacy/ClearBrowsingDataPreferencesTest.java?l=428

i.e.
WebContentsImpl::DocumentOnLoadCompleted() is called
https://cs.chromium.org/chromium/src/content/browser/web_contents/web_contents_impl.cc?l=5289

which consequently sets current_page_quality_ to PageQuality::HIGH:
https://cs.chromium.org/chromium/src/components/offline_pages/core/snapshot_controller.cc?l=161

so that once a dialog is shown and OnHide() is called this check in recent_tab_helper.cc is not satisfied:
https://cs.chromium.org/chromium/src/chrome/browser/offline_pages/recent_tab_helper.cc?l=324

and last_n snapshot is created.

If the snapshot is created, in the next loadUrl() call:
https://cs.chromium.org/chromium/src/chrome/android/javatests/src/org/chromium/chrome/browser/preferences/privacy/ClearBrowsingDataPreferencesTest.java?l=435

attempt to load an offline page will be made:
https://cs.chromium.org/chromium/src/chrome/browser/offline_pages/offline_page_request_job.cc?l=919

but it will fail with result = -1 for file_path = /data/user/0/org.chromium.chrome/cache/Offline Pages/archives/7c482bec-4e72-4345-ae0e-025e5f845a3f.mhtml