syncer: exit broadcast as soon as possible

[rleungx]

What problem does this PR solve?

Issue Number: Close #9017

What is changed and how does it work?

Check List

Tests

• Unit test

Release note

None.

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[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 76.27%. Comparing base (6cadbe7) to head (3f64d88).
Report is 1 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #9018      +/-   ##
==========================================
+ Coverage   76.25%   76.27%   +0.02%     
==========================================
  Files         466      466              
  Lines       70733    70740       +7     
==========================================
+ Hits        53938    53959      +21     
+ Misses      13437    13423      -14     
  Partials     3358     3358              

Flag	Coverage Δ
unittests	76.27% <100.00%> (+0.02%)	⬆️

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[nolouch]

Can this done channel be defined in the function scope?