[PR #8680/4f41d05 backport][3.11] Fix close race that prevented the close code from reaching the client

[bdraco]

(cherry picked from commit 4f41d05)

[codecov]

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Additional details and impacted files

@@           Coverage Diff           @@
##             3.11    #8687   +/-   ##
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  Coverage        ?   97.70%           
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  Files           ?      108           
  Lines           ?    34050           
  Branches        ?     4048           
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  Hits            ?    33268           
  Misses          ?      585           
  Partials        ?      197           

Flag	Coverage Δ
CI-GHA	97.60% <100.00%> (?)
OS-Linux	97.28% <100.00%> (?)
OS-Windows	94.80% <100.00%> (?)
OS-macOS	96.94% <100.00%> (?)
Py-3.10.11	97.04% <100.00%> (?)
Py-3.10.14	96.99% <100.00%> (?)
Py-3.11.9	97.23% <100.00%> (?)
Py-3.12.4	97.33% <100.00%> (?)
Py-3.8.10	94.42% <100.00%> (?)
Py-3.8.18	96.77% <100.00%> (?)
Py-3.9.13	96.92% <97.82%> (?)
Py-3.9.19	96.88% <97.82%> (?)
Py-pypy7.3.16	96.47% <100.00%> (?)
VM-macos	96.94% <100.00%> (?)
VM-ubuntu	97.28% <100.00%> (?)
VM-windows	94.80% <100.00%> (?)

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