Fix stale conversation sync labels on mobile

tests/conversation-home-items.test.ts

@@ -818,6 +818,43 @@ test("conversation items prefer latest observed codex activity over stale last m
   assert.equal(thread?.latestReplyLabel, formatTimestampLabel("2026-04-04T11:48:00+08:00"));
 });
 
+test("conversation items mark stale context-backed timestamps as waiting for sync", async () => {
+  await setup();
+  const state = await readState();
+  state.projects = state.projects.filter((project) => project.id === "master-agent");
+  state.projects.push({
+    ...buildImportedThreadProject(
+      "mac-studio",
+      "stale-context-project",
+      "Boss",
+      "boss",
+      "上下文老化线程",
+      "thread-stale",
+      "2026-03-25T10:58:00+08:00",
+    ),
+    preview: "老上下文还在挂着",
+    unreadCount: 1,
+  });
+  state.threadContextSnapshots.push(
+    buildThreadContextSnapshot(
+      "stale-context-project",
+      "thread-stale",
+      "上下文老化线程",
+      "watch",
+      false,
+      58,
+      "2026-03-25T10:58:00+08:00",
+      "mac-studio",
+    ),
+  );
+
+  const item = getConversationHomeItems(state).find((entry) => entry.projectId === "stale-context-project");
+
+  assert.ok(item, "expected stale context conversation");
+  assert.equal(item?.contextBudgetIndicator.visible, true);
+  assert.equal(item?.latestReplyLabel, "待同步");
+});
+
 test("default seeded conversations no longer expose Boss 移动控制台", async () => {
   await setup();
   const state = await readState();