[TheGreatAxios]

Closes CL-11526

Why
Catch-up, Triage again and the Settings status all looked for a live pr-triage deployment, even in a workspace whose pull requests run in managers. Without that deployment, catch-up stopped, Triage again answered "not running" and Settings said Triage was off, so the old workers could not be removed.

What changed

• While a host's listener is live, the catch-up pass runs without a pr-triage deployment. It neither replaces nor redeploys one, and it mails its heads through the listener as webhooks already do.
• With the listener live, Triage again needs no pr-triage deployment and goes through the listener. Whether the head is already being triaged comes from its manager, as recorded by fix(manager): record the manager's verdict when it reports done #637, which this PR is stacked on.
• Settings counts a host's listener as Triage running only while the hub counts it as up. The existing workflow-versions response now gives each deployment a live field for this. No new route.
• A listener that was deployed but never got a sidecar within two minutes is released, the same way a stuck manager is, so the next Upgrade deploys it again.
• Old pr-triage deployments that are still running are left alone while a listener is live. The CL-11525 removal slices cancel them.
• If the old pr-triage deployment was being replaced when the listener came up, that replacement is abandoned and its copy keeps running until the CL-11525 removal slices cancel it.
• A listener counts as live the same way a manager does: it is running, it is on its sidecar, or it was deployed in the last two minutes. Catch-up, Triage again and incoming mail all use this one check.
• A workspace without a listener works as before. Triage again is GitHub-only today, so Bitbucket has no route to change.
• ARCHITECTURE.md (reconciler) and the route's OpenAPI text describe the listener case.

Verified

• New tests: the reconciler catches up a head with only a listener live, and mails it only once over many passes while its manager runs and after it reports (with the manager's runs taken away, it mails 3 more times). Triage again mails with only a listener live and refuses a head its manager is running. Both failed before the change and pass after. A third test checks that a listener with no sidecar after two minutes does not count as live and is released.
• Hub and web tests: 766 of 767 pass in the full run. The one failure, in triage-verdicts.postgres.test.ts (a file this change does not touch), was a 5-second timeout while the machine was heavily loaded, and that file passes 5 of 5 on its own.
• bun run check, bun run lint and the policy check pass.

[linear-code]

CL-11526

[corbits-triage]

Triage: Pass

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