Show Codex background task counts

PLAN.md

@@ -869,7 +869,7 @@ the reader was not asking after, and one of them turned out to be a whole shell
 command drawn as centred prose, because its words came from somewhere with no
 reason to keep them short. The parent's transcript gets a row for a message a
 subagent genuinely *sends* it, which arrives by the peer path and already has
-one. The live provider-reported count beside the parent session's status is
+one. The live measured count beside the parent session's status is
 deliberately the only background-task UI until there is a design for inspecting
 them.
 
@@ -899,6 +899,13 @@ The ordinary notifications still supply outcomes and summaries, including when
 one is ordered after the level already corrected the status. Older CLIs send no
 level and retain the edge fallback below.
 
+Codex has no equivalent level for the whole session. Its collaboration
+lifecycle does have the fact the screen needs: the subagent registry counts
+its open child threads, including ones found on disk after adoption, and emits
+the same `backgroundTasks` event whenever that count changes. Backgrounded
+Codex commands are not included because app-server exposes no session-level
+set for them; the UI does not infer a count from tool cards.
+
 What the notification is still used for is the status: it is what closes a task
 in `Status::Waiting`'s bookkeeping. Handled once, however many of the two
 lifecycle shapes (`task_notification`, `task_updated`) arrive — whichever gets