fix(chat): keep mention searches open and align resource menus #33900

Job	Run time
Test and Build / PostgreSQL integration (push)	9m 7s
Test and Build / PostgreSQL integration (migrate)	8m 49s
Test and Build / Lint and Test	5m 22s
Test and Build / Build App	4m 9s
Detect Desktop Changes	0s
Build Dev ECR	0s
Prepare Trigger.dev	0s
Check Docs Changes	0s
Build ARM64 (GHCR Only)	0s
Migrate Dev DB	0s
Build AMD64	0s
Create Desktop Prerelease	0s
Check Desktop Signing Secrets	0s
Detect Version	0s
Publish Desktop Prerelease	0s
Desktop Prerelease Build	0s
Prune Desktop Prereleases	0s
Migrate DB	-1s
Promote Images	0s
Promote Trigger.dev	0s
Process Docs	0s
Create GHCR Manifests	0s
Resolve Attestation Subjects	0s
Attest Images	0s
Create GitHub Release	0s
Desktop Release	0s
	27m 26s

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