Pass on information about the embedder state using the fact that tasks
are run from top level

Bug: chromium:893944
Change-Id: I01441778770c5acc784540e496eec5c3fdb87796
Reviewed-on: https://chromium-review.googlesource.com/c/1273048
Commit-Queue: Michael Lippautz <mlippautz@chromium.org>
Reviewed-by: Ulan Degenbaev <ulan@chromium.org>
Cr-Commit-Position: refs/heads/master@{#56540}