Show live background task counts

PLAN.md

@@ -858,7 +858,9 @@ 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.
+one. The live provider-reported count beside the parent session's status is
+deliberately the only background-task UI until there is a design for inspecting
+them.
 
 **The report goes to whichever record is the only one of it**, and the two cases
 are different places. A subagent has a transcript of its own, and its closing
@@ -874,7 +876,10 @@ the ones a stale "running in background" reads worst on.
 (2026-09-15). Claude Code 2.1.261 added
 `background_tasks_changed { tasks: [...] }` with replace semantics expressly so
 a missed bookend cannot wedge a running indicator. Its ids are not correlated
-with the edge stream; this side uses the authoritative empty/nonempty level.
+with the edge stream; this side uses the authoritative empty/nonempty level and
+its measured size. That size is exposed as `backgroundTasks` in the session row
+and event stream, and is drawn beside the status; background tasks do not become
+subagent cards.
 The driver sends a repeated `initialize` when it adopts a CLI, which prompts a
 full snapshot without restarting the conversation. A parent already recorded as idle or waiting can
 apply it immediately; one adopted mid-turn waits for the result boundary,