[FEATURE] PR monitor: detect merge conflicts and wake the session to resolve, not just CI failures and comments

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

The desktop PR monitor polls check results and wakes Claude on CI failures and review comments — but not when the PR becomes conflicted because the base branch advanced. GitHub emits no CI event for this, so in multi-PR workflows (several sessions' PRs merging in sequence) every merge silently strands the sibling PRs in a conflicted state until a human notices. Users cover the gap with hand-rolled background loops (git fetch origin main && git merge origin/main, resolve, re-test, push) that duplicate polling the harness is already doing.

Proposed Solution

Since the monitor already polls via the gh CLI, include mergeability (mergeable / mergeStateStatus) in the poll. When the PR transitions to CONFLICTING, wake the session to merge the base branch back in (or rebase, per a setting), resolve, re-run the local gate, and push — governed by the same Auto-fix toggle (or a sub-toggle alongside it).

Alternative Solutions

A background agent loop polling and merging on an interval — works, but burns session attention and tokens to duplicate a poll the CI monitor already performs.

Priority

Medium - Would be very helpful

Feature Category

Other

Use Case Example

1. A dispatcher and N worker sessions each own an open PR against main.
2. PR 1 merges; PRs 2..N are now conflicting, but nothing wakes their sessions.
3. With conflict-aware wake-ups, each worker heals its own PR as soon as the base advances.
4. Today each worker needs a hand-rolled watcher loop for exactly this case.

Additional Context

Sibling request #66097 and bug #68083 concern the same CI monitor; this one only extends what the existing poll watches, not how it acts on comments.