[DevTools] fix: keep search query in a local sync state (#34423) #1386

Job	Run time
Lint babel-plugin-react-compiler	20s
Jest babel-plugin-react-compiler	48s
Discover yarn workspaces	6s
Test react-compiler-runtime	1s
Test react-compiler-healthcheck	1s
Test snap	1s
Test make-read-only-util	1d 0h 0m 0s
Test babel-plugin-react-compiler	1d 0h 0m 0s
Test eslint-plugin-react-compiler	1s
Test react-mcp-server	1s
Test react-forgive	1s
	2d 0h 1m 20s

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