fix(github): stop PR-status requests from starving startup connection pool

Watching N worktrees fired N PR-status requests at once (startWatching
called refresh() directly, bypassing the batch limiter). Each request can
take 20s+ under GitHub secondary-rate-limiting, and N of them saturate the
browser's ~6 HTTP/1.1 connections per origin, starving the critical path
(bootstrap session.status, diffs, sending messages) until they finish — the
UI appeared frozen for ~20s on startup.

- Gate all PR-status network calls through a global concurrency semaphore
  (max 2), so free sockets always remain for critical traffic.
- Bound resolveGitHubPrStatus with a 12s timeout so a slow request fails
  fast instead of holding a socket; the client keeps its last-known status.
- Reuse already-fetched repo metadata for the default branch instead of a
  redundant repos.get, reducing serial GitHub calls (less rate-limiting).

packages/ui/src/stores/useGitHubPrStatusStore.ts

@@ -93,6 +93,43 @@ const timers = new Map<string, number>();
 const bootstrapTimers = new Map<string, number[]>();
 const inFlightBySignature = new Set<string>();
 const lastRefreshBySignature = new Map<string, number>();
+
+// Global concurrency gate for PR-status network requests.
+//
+// PR status is non-critical chrome, but each request can be slow (the server
+// makes many serial GitHub API calls and GitHub secondary-rate-limits bursts,
+// so a single request can take 20s+). The browser allows only ~6 concurrent
+// HTTP/1.1 connections per origin. Without this cap, watching N worktrees fires
+// N PR-status requests at once (each startWatching() calls refresh() directly,
+// bypassing refreshTargets' batch limiter), which saturates the connection pool
+// and starves the critical path (bootstrap session.status, diffs, sending
+// messages) for the full duration — the whole UI appears frozen on startup.
+//
+// Capping concurrency low guarantees free sockets remain for critical traffic.
+const PR_STATUS_NETWORK_CONCURRENCY = 2;
+let prStatusNetworkActive = 0;
+const prStatusNetworkWaiters: Array<() => void> = [];
+
+const acquirePrStatusNetworkSlot = (): Promise<void> => {
+  if (prStatusNetworkActive < PR_STATUS_NETWORK_CONCURRENCY) {
+    prStatusNetworkActive += 1;
+    return Promise.resolve();
+  }
+  return new Promise<void>((resolve) => {
+    prStatusNetworkWaiters.push(resolve);
+  });
+};
+
+const releasePrStatusNetworkSlot = (): void => {
+  const next = prStatusNetworkWaiters.shift();
+  if (next) {
+    // Hand the slot directly to the next waiter — keep the active count steady.
+    next();
+    return;
+  }
+  prStatusNetworkActive = Math.max(0, prStatusNetworkActive - 1);
+};
+
 const createEntry = (): PrStatusEntry => ({
   status: null,
   isLoading: false,
@@ -488,7 +525,13 @@ export const useGitHubPrStatusStore = create<GitHubPrStatusStore>()(
             activeRequestCount: prev.activeRequestCount + 1,
             totalRequestCount: prev.totalRequestCount + 1,
           }));
-          const next = await params.github.prStatus(params.directory, params.branch, params.remoteName ?? undefined, { force: options?.force });
+          await acquirePrStatusNetworkSlot();
+          let next: GitHubPullRequestStatus;
+          try {
+            next = await params.github.prStatus(params.directory, params.branch, params.remoteName ?? undefined, { force: options?.force });
+          } finally {
+            releasePrStatusNetworkSlot();
+          }
           set((prev) => {
             const nextEntries = { ...prev.entries };
             signatureKeys.forEach((signatureKey) => {