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This patch affects tests only.

We want to cache the |network_context_request_| to avoid triggering
connection error handler infinitely.

(Also organized unrelated BackgroundFetchBrowserTest.* tests)

Bug: 889965
Change-Id: I9fc67b03f19b8a9575db7714ea57ec41fc0a7711
Reviewed-on: https://chromium-review.googlesource.com/1249727
Commit-Queue: Chong Zhang <chongz@chromium.org>
Reviewed-by: Matt Menke <mmenke@chromium.org>
Cr-Commit-Position: refs/heads/master@{#594946}