Tighten mention search loading-state detection

[ZhangHanDong]

The current mention search starts as soon as room_members is non-empty, even though the SDK may still be streaming the rest of the members. Create an issue to track improving the loading heuristics (e.g., compare against active_members_count() or use incremental updates) so the popup waits for the full list—or at least a clearly defined threshold—before firing the search.

[kevinaboos]

I don't think we should wait at all, and certainly not for the full list to be received. Starting the search as soon as there is ≥1 room member is best because it gives the fastest results.

Incremental updates is the way to go; I assumed you were already doing that in #614 (but i haven't investigated yet).

[ZhangHanDong]

@kevinaboos Yes. The widget kicks off the background search as soon as the cached member vector is non-empty in #614 .