[Bug]: Thread with an open PR auto-settles after every completed turn — even with background watchers still running (regression, started 2026-08-06)

Before submitting

• I searched existing issues and did not find a duplicate.
• I included enough detail to reproduce or investigate the problem.

Area

Not sure

Steps to reproduce

1. Start a thread; the agent's first completed turn creates a PR (still open, neither merged nor closed — the thread shows the PR badge).
2. Continue working in the thread: every time a turn completes, the thread immediately auto-settles.
3. This happens even when the turn explicitly leaves background work running — e.g. the agent's last message says it is waiting on a background watcher for a PR re-review, yet the "This thread is settled" banner appears right away.

Observed on the desktop app (macOS).

Expected behavior

A thread on an open PR should not auto-settle. At minimum, a thread with active background commands/watchers should never settle.

Actual behavior

Once a turn has created a PR, every subsequent completed turn settles the thread. I have to click "Un-settle" (or send a message) after every single turn, which makes the PR-follow-up workflow miserable.

Additional context

Started this morning (2026-08-06) — was not happening before, so it looks like a recent nightly regression. Not a duplicate of #4982: that one asks for a toggle on the merged/closed-PR auto-settle rule; here the PR is still open. Possibly related to the settle-state issues in #4817 / #5377, but neither describes this trigger.

Start by reproducing the desktop app flow described: create an open PR, continue the thread, and leave a background watcher running. Trace the thread auto-settle entry point and compare its handling of open PRs and active background commands; done means the thread stays unsettled until the PR is merged or closed and no active watcher is running.