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TSAN reported lock-order-inversion because we were using two mutexes. Simplify the code to use just one.

Bug: 915200
Change-Id: I5dbda4ca21cd4f5fa95ecb1c9b51922262798169
Reviewed-on: https://chromium-review.googlesource.com/c/1384384
Commit-Queue: Keishi Hattori <keishi@chromium.org>
Reviewed-by: Kentaro Hara <haraken@chromium.org>
Reviewed-by: Michael Lippautz <mlippautz@chromium.org>
Cr-Commit-Position: refs/heads/master@{#618754}