https://chromium-review.googlesource.com/c/chromium/src/+/1114921 has an approach. Writing tests is a bit painful.

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/36985c8e090f74799ba1d181b008d4641c0f7146

commit 36985c8e090f74799ba1d181b008d4641c0f7146
Author: Josh Karlin <jkarlin@chromium.org>
Date: Wed Jun 27 15:29:57 2018

Prioritize replies as well as tasks

What:
PrioritizedTaskRunner currently runs its tasks (from RunTaskAndReply) in prioritized order but not the replies. This CL adds a priority queue for replies to run in order as well.

Why:
At least for the cache, the reply task is equally important to the posted task and should run in priority order.


Bug:  856674 
Change-Id: Icea37adb96891473292e0c38805c42e84f9d4f5f
Reviewed-on: https://chromium-review.googlesource.com/1114921
Reviewed-by: Maks Orlovich <morlovich@chromium.org>
Commit-Queue: Josh Karlin <jkarlin@chromium.org>
Cr-Commit-Position: refs/heads/master@{#570781}
[modify] https://crrev.com/36985c8e090f74799ba1d181b008d4641c0f7146/net/base/prioritized_task_runner.cc
[modify] https://crrev.com/36985c8e090f74799ba1d181b008d4641c0f7146/net/base/prioritized_task_runner.h
[modify] https://crrev.com/36985c8e090f74799ba1d181b008d4641c0f7146/net/base/prioritized_task_runner_unittest.cc