Show live background task counts

AGENTS.md

@@ -264,7 +264,10 @@ written, and the fold uses that same predicate to decide a reply is settled.
 - **Claude's background-task level is authority; two edge sources are the
   fallback.** Since Claude Code 2.1.261,
   `background_tasks_changed { tasks: [...] }` replaces the live set and repairs
-  a missed ending edge. An adopted CLI is sent a repeated `initialize` to ask
+  a missed ending edge. Its array size is also the measured `backgroundTasks`
+  count exposed on the session row and event stream; the phone draws a nonzero
+  count beside the status rather than deriving one from `waiting` or from the
+  subagent directory. An adopted CLI is sent a repeated `initialize` to ask
   for the current set. Reconcile only between turns or at a result boundary:
   a foreground agent is legitimately absent from a background-only snapshot.
   Older CLIs still need both edge sources: `open_tasks` knows about a