A run outlives the connection that started it
When you send a message, the deployment starts a turn and your screen attaches to it as a viewer. Close the tab, lock the phone, switch apps, walk from Wi-Fi onto mobile data: the viewer goes away and the turn does not notice. The agent keeps reading, calling tools and writing, because none of that was happening in your browser.
Come back - the same tab, a new one, or another device - and the screen rejoins the reply exactly where you stopped seeing it. It fills in the words, the tool steps and the files you missed, then carries on live. A reply that finished while you were away is simply there. A conversation the agent is still working on shows as busy in the list on every device, including one that never saw the turn start, and the working indicator names the surface the turn is running from, so a laptop can watch a reply that began on Telegram.
Two consequences follow. A conversation runs one reply at a time, so sending again from your phone while the laptop's turn is still running does not start a second one; the message either steers the reply in progress or waits its turn. And two status lines mean different things. Reconnecting means the reply is catching up and nothing was lost, so there is nothing to resend. Connection lost means what it says - the reply is genuinely gone, because the deployment restarted or the reply ended long enough ago to have been cleared - and that is when a Retry button appears.
Stop means stop
Because closing a window ends nothing, Stop is the control that does. It tells the deployment to end the turn, which is why it works from any device: you can stop from your phone a task you started on a laptop that is now shut in a bag, whether or not the tab that started it is still open. Closing the tab is not a stop.