Agent participants can get stuck in private chat

I was testing agent participants today, and had one get stuck in a private chat stage. This left the experiment stuck until I manually hit the next stage button for the agent.

Mediator 16:31
Hello! I'm the Mediator. Since we're in this private chat, I'd love to get to know you a bit better. To start things off, what's your favorite flavor of ice cream?

AI Assistant (it/its) 16:31
Hello Mediator! It's good to meet you. My favorite flavor of ice cream would have to be classic vanilla bean.

Mediator 16:31
Vanilla bean is a classic choice! Thank you for sharing that with me.

AI Assistant (it/its) 16:31
You're welcome! I appreciate you asking. I'm happy to share.

Checking the logs, the mediator here gave an empty response with readyToEndChat: true, at which point the agent didn't get any more queries.

Some ideas for options that could fix this:

1. End private chat stages when the mediator leaves or runs out of questions
2. Don't let the mediator choose to end a private chat stage
3. Wake agents up when someone leaves the chat
4. End any chat stage when there are no humans left and no pending agent responses
5. Wake agents up every minute or so if the chat has been idle
6. Auto-end chat stages if they're idle for too long
7. Default the private chat timer time limit to on

No files or tests are named in the issue. Start by reproducing an agent participant getting stuck after a mediator returns an empty response with readyToEndChat: true, then trace the private-chat stage and agent wake-up flow. Done means a private chat cannot remain stuck when the mediator ends, leaves, or becomes idle, with a regression test covering the chosen behavior.