+yhanada, IIRC, you investigated this?

There are two reasons for killing the message center immediately after showing the virtual keyboard. The first reason is that the widget containing the message center is configured to close after deactivating (see WebNotificationBubbleWrapper in web_notification_tray.cc). The second reason is WebNotificationTray closes the message center when a user clicks outside of the message center (see WebNotificationTray::ClickedOutsideBubble).

I haven't come up with a smart solution for it yet.

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5

commit cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5
Author: yhanada <yhanada@chromium.org>
Date: Fri Jul 14 10:53:12 2017

Do not close the message center when tapping on the virtual keyboard.

Android notifications from ARC++ implement inline reply feature, so
it needs to be able to open the virtual keyboard and the message center
simultaneously.

- Pass the hit NativeView instead of the hit Widget to OnPointerEventObserved.
- Do not call ClickedOutsideBubble when clicking on the virtual
  keyboard.
    
Bug:  722248 
Test: Unit tests are added.

Change-Id: I7df8a1ec6cf2c1af4a5e7274cb93c8af6b3e1022
Reviewed-on: https://chromium-review.googlesource.com/532768
Commit-Queue: Yuichiro Hanada <yhanada@chromium.org>
Reviewed-by: James Cook <jamescook@chromium.org>
Reviewed-by: Sadrul Chowdhury <sadrul@chromium.org>
Cr-Commit-Position: refs/heads/master@{#486724}
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/BUILD.gn
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/aura/pointer_watcher_adapter.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/aura/pointer_watcher_adapter.h
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/aura/pointer_watcher_adapter_unittest.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/autoclick/mus/autoclick_application.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/metrics/pointer_metrics_recorder.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/metrics/pointer_metrics_recorder.h
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/metrics/pointer_metrics_recorder_unittest.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/public/cpp/immersive/immersive_fullscreen_controller.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/public/cpp/immersive/immersive_fullscreen_controller.h
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/shelf/overflow_bubble.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/shelf/overflow_bubble.h
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/shelf/shelf_tooltip_manager.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/shelf/shelf_tooltip_manager.h
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/system/tray/tray_bubble_wrapper.h
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/system/tray/tray_event_filter.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/system/tray/tray_event_filter.h
[add] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/system/tray/tray_event_filter_unittest.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ash/touch_hud/mus/touch_hud_application.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/chrome/browser/ui/ash/app_list/app_list_presenter_delegate_mus.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/chrome/browser/ui/ash/app_list/app_list_presenter_delegate_mus.h
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ui/views/mus/pointer_watcher_event_router.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ui/views/mus/pointer_watcher_event_router_unittest.cc
[modify] https://crrev.com/cfee1fb0708b8b5ce1a26e7365b7f296318ae5c5/ui/views/pointer_watcher.h

Woohoo! Thank you