Auto-snooze green PRs from the PR watch tool, opt-in setting

[flamboh]

Recently, a tool for agents to asynchronously watch PRs was added. Once a PR is clean and has no failing CI or comments, the thread returns to the inbox. It'd be nice if we could have them auto-snoozed so that they stay out of the way while keeping it clear they're not fully done yet.

below is claude's full description:

The problem

With [#15057](#15057), an agent can watch its pull request and gets woken when checks fail, required checks pass, someone reviews, or the branch conflicts. That takes care of the agent. The thread itself still sits in the sidebar, though, even when there's nothing for me to do. Often a green PR is just waiting on a human reviewer, and that can take hours or days.

If I run many agents at once, those threads add up. Every green PR waiting on review looks the same in the sidebar as a thread that actually needs me, so I end up snoozing them by hand and remembering to come back.

The idea

Add an opt-in setting, something like Settings → Snooze green pull requests. When a thread's watched pull request goes green:

1. The agent still gets its normal green wake and finishes whatever it does with it.
2. Once the thread is idle, T3 Code snoozes it.
3. The thread un-snoozes when the watch has real news: a check fails, someone outside leaves review feedback, or the branch starts to conflict. The agent wakes with that news as it does today, and the thread comes back to the top.

So a thread only asks for my attention when its PR needs something.

Details

• Off by default. With the setting off, nothing changes.
• Snoozing doesn't decide anything about merging. It only affects what I see. The agent still gets every wake.
• Manual un-snooze wins. If I wake the thread myself, it stays awake for that green state. A new push that goes green again can snooze it again.
• The watch keeps running while the thread is snoozed, so the thread can always wake up again.
• Merging or closing settles it. When a snoozed thread's PR merges or closes, the work is done, so the thread settles instead of staying snoozed.
• Works with every provider and on every surface. The server decides when to snooze, so web, desktop and mobile all behave the same. The setting appears in web and mobile settings.
• Small change. It reuses the existing watch state, snooze and settle. No new background service.

Open questions

• Is "snooze until something happens" enough, or would people want a time limit, like "snooze for at most 24h"?
• Should the sidebar show why a snoozed thread woke up (failed check, review, conflict), or is the agent's wake message in the transcript enough?