The stream catch-up mode was represented by two fields that could drift:
the private catchupStreamSessionId guard (which suppresses live transcript
events while the in-flight message prefix is missing) and the public
isReceivingPartialStream flag (which drives the badge). The socket reconnect
path (refreshSelectedSession) set the flag from status without updating the
guard, so a later idle status.update was skipped by the guard check and the
badge never cleared.
Route every mutation of the mode through a single setStreamCatchup helper so
the guard and badge can't disagree, and clear the mode on any non-streaming
status for the selected session.
