[PR #8680/4f41d05 backport][3.10] Fix close race that prevented the close code from reaching the client

[bdraco]

(cherry picked from commit 4f41d05)

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.70%. Comparing base (8c686e6) to head (13db6d5).
Report is 210 commits behind head on 3.10.

Additional details and impacted files

@@            Coverage Diff             @@
##             3.10    #8686      +/-   ##
==========================================
- Coverage   97.72%   97.70%   -0.03%     
==========================================
  Files         108      108              
  Lines       34015    34050      +35     
  Branches     4047     4048       +1     
==========================================
+ Hits        33241    33268      +27     
- Misses        577      585       +8     
  Partials      197      197              

Flag	Coverage Δ
CI-GHA	97.60% <100.00%> (-0.03%)	⬇️
OS-Linux	97.28% <100.00%> (-0.02%)	⬇️
OS-Windows	94.80% <100.00%> (-0.02%)	⬇️
OS-macOS	96.94% <100.00%> (-0.03%)	⬇️
Py-3.10.11	97.04% <100.00%> (+2.42%)	⬆️
Py-3.10.14	96.99% <100.00%> (-0.03%)	⬇️
Py-3.11.9	97.23% <100.00%> (-0.03%)	⬇️
Py-3.12.4	97.33% <100.00%> (-0.03%)	⬇️
Py-3.8.10	94.42% <100.00%> (-0.02%)	⬇️
Py-3.8.18	96.77% <100.00%> (-0.03%)	⬇️
Py-3.9.13	96.92% <97.82%> (-0.03%)	⬇️
Py-3.9.19	96.78% <97.82%> (-0.13%)	⬇️
Py-pypy7.3.16	96.46% <100.00%> (-0.03%)	⬇️
VM-macos	96.94% <100.00%> (-0.03%)	⬇️
VM-ubuntu	97.28% <100.00%> (-0.02%)	⬇️
VM-windows	94.80% <100.00%> (-0.02%)	⬇️

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