Contact Thread support and request:

"We'd like a dedicated 'Listening Mode inactivity timeout' event trigger for Flows. Currently the only way to approximate this is filtering on the AI New status change, but this fires on every transition to AI New — not just when Listening Mode times out due to inactivity. A specific trigger for this event would allow us to send chat-only inactivity messages without false positives on escalated or re-triaged tickets."