Enables or disables GitHub auto-merge (squash) for this session’s pull request — the same identity the GET resolves, from the owning run or the session branch — using an installation token clamped to the owning agent’s repository tier — the write requires that clamp to actually carry pull_requests: write, so a read- or comment-tier agent is refused (403) rather than escalated. Idempotent: asking for the state the PR is already in succeeds without a mutation. 404 mirrors the GET; 409 relays GitHub declining the state change (for example a pull request whose checks already pass, which GitHub arms nothing for); 429 is GitHub rate limiting; 502 is GitHub unreachable.