fix(desktop): sidebar status indicators lag for background sessions

The sidebar working dot didn't update for background sessions until the
user opened them. Two coupled causes:

1. The gateway's session.info event payload omitted stored_session_id,
   so the desktop app had no way to map a background session's runtime
   id to its stored id. Without the stored id, setSessionWorking(null,
   ...) was a no-op — the $workingSessionIds atom never updated.

2. The running→busy transition in the session.info handler was gated on
   `apply` (active session only). The gate correctly scopes view-only
   side effects (setCurrentModel, setCurrentCwd, etc.) to the focused
   chat, but the per-session busy state drives the sidebar indicator and
   must reach every session. updateSessionState only mutates the
   per-runtime cache entry, and syncSessionStateToView already guards
   the view publish to the active session, so ungating is safe.

Fix: add stored_session_id to _session_info() in tui_gateway/server.py,
add the field to GatewayEventPayload, pass it to updateSessionState in
the session.info handler, and ungate the running→busy transition.

apps/desktop/src/app/chat/sidebar/session-row.tsx

@@ -313,10 +313,11 @@ const DOT_VARIANTS: Record<SessionDotState, DotVariant> = {
     role: 'status'
   },
   // Pulsing gray — a terminal(background=true) process is alive while the LLM
-  // is idle. Gray (not accent) reads as "something chugging along".
+  // is idle. Gray (not accent) reads as "something chugging along". Brighter
+  // than muted-foreground so it's visible against the sidebar surface.
   background: {
     ariaLabel: r => r.backgroundRunning,
-    className: `${DOT_BASE} bg-muted-foreground/50 ${PING} before:bg-muted-foreground/50 before:opacity-50`,
+    className: `${DOT_BASE} bg-muted-foreground/80 ${PING} before:bg-muted-foreground/80 before:opacity-60`,
     role: 'status',
     title: r => r.backgroundRunning
   },

apps/desktop/src/app/session/hooks/use-message-stream/gateway-event.ts

@@ -218,17 +218,30 @@ export function useGatewayEventHandler(deps: GatewayEventDeps) {
         }
 
         if (sessionId && hasStatePatch) {
-          updateSessionState(sessionId, state => ({
-            ...state,
-            ...statePatch,
-            branch: statePatch.branch ?? state.branch,
-            cwd: statePatch.cwd ?? state.cwd
-          }))
+          updateSessionState(
+            sessionId,
+            state => ({
+              ...state,
+              ...statePatch,
+              branch: statePatch.branch ?? state.branch,
+              cwd: statePatch.cwd ?? state.cwd
+            }),
+            payload?.stored_session_id || undefined
+          )
         }
 
-        if (apply) {
-          if (runningChanged && sessionId) {
-            updateSessionState(sessionId, state => {
+        // The running→busy transition must reach EVERY session, not just the
+        // active one. The `apply` gate above correctly scopes view-only side
+        // effects (setCurrentModel, setCurrentCwd, etc.) to the focused chat,
+        // but the per-session busy state is what drives the sidebar working
+        // indicator — a background session's turn start/finish must update
+        // its dot without the user opening it. updateSessionState only
+        // mutates the per-runtime cache entry, and syncSessionStateToView
+        // guards the view publish to the active session, so this is safe.
+        if (runningChanged && sessionId) {
+          updateSessionState(
+            sessionId,
+            state => {
               const busy = Boolean(payload!.running)
 
               if (state.busy === busy && (busy || !state.awaitingResponse)) {
@@ -255,8 +268,9 @@ export function useGatewayEventHandler(deps: GatewayEventDeps) {
                 streamId: null,
                 turnStartedAt: null
               }
-            })
-          }
+            },
+            payload?.stored_session_id || undefined
+          )
         }
 
         if (payload?.usage && (!explicitSid || isActiveEvent)) {

apps/desktop/src/lib/chat-messages.ts

@@ -79,6 +79,10 @@ export type GatewayEventPayload = {
   // session.title (live auto-title push) — stored session id + generated title
   session_id?: string
   title?: string
+  // session.info — the stored (durable) session id for this runtime session.
+  // Lets the desktop app map runtime→stored for background sessions it hasn't
+  // opened, so the sidebar working indicator updates without opening the chat.
+  stored_session_id?: string
   // moa.reference / moa.aggregating (Mixture of Agents per-model relay)
   label?: string
   index?: number

tui_gateway/server.py

@@ -3656,6 +3656,7 @@ def _session_info(agent, session: dict | None = None) -> dict:
         "personality": str(personality or ""),
         "running": bool((session or {}).get("running")),
         "title": _session_live_title(session or {}, session_key) if session_key else "",
+        "stored_session_id": session_key or "",
         "desktop_contract": DESKTOP_BACKEND_CONTRACT,
         "version": "",
         "release_date": "",