Remove unnecessary check in `start-proxy-action-post` #11184

Job	Run time
Split workflow (macos-latest, default)	2m 31s
Split workflow (macos-latest, linked)	1m 56s
Split workflow (ubuntu-latest, linked)	1m 7s
Split workflow (ubuntu-latest, nightly-latest)	1m 18s
Split workflow (ubuntu-latest, default)	1m 10s
Split workflow (macos-latest, nightly-latest)	1m 28s
	9m 30s

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