Merge branch 'perf/lag': dashboard lag fixes

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

# Conflicts:
#	client/src/components/Sagi/PetSprite.tsx
#	client/src/components/Sagi/useSagiBrain.ts

server/routes/lanes.js

@@ -211,7 +211,11 @@ function featurePayload(feature) {
 /** Push the current state of one lane to every connected client. */
 function broadcastLane(id) {
   const lane = lanesLib.getLane(id);
-  if (lane) broadcast("lane_update", { lane: payload(lane) });
+  if (!lane) return;
+  // The cached list-sessions (as GET / uses), not a synchronous has-session —
+  // through sudo for a member's pane — on every hook that moves a lane.
+  const live = lane.run_id ? tmux.liveSessions() : null;
+  broadcast("lane_update", { lane: payload(lane, live) });
 }
 
 router.get("/", (req, res) => {