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"@jmfederico/pi-web": patch
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Fix the "Catching up…" badge sometimes staying visible after a session goes idle. The stream catch-up mode was tracked by two fields that could drift — a private guard and the public badge flag — and the socket reconnect path updated one without the other, so the terminating idle status no longer cleared the badge. Both facets now route through a single source of truth, and any idle status for the selected session reliably dismisses the badge.