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This is only ever used to track if the dropdown is pressed; replace it
with a bool.

Also removes SchedulePaint() calls around changing the dropdown image,
which should be done automatically and thus are redundant.

Bug: none
Change-Id: I457db5f1bc953d0c97cc51b77a13c433d80c6fd4
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2305373
Commit-Queue: Peter Kasting <pkasting@chromium.org>
Reviewed-by: Xing Liu <xingliu@chromium.org>
Cr-Commit-Position: refs/heads/master@{#790201}