feat: let a background task ask the chat that started it, not just the Inbox

An async sub-agent runs in a session of its own, so the rich per-session events
that draw the inline approval card never reach the chat's socket — only the
id-only inbox lifecycle ones do. A task blocked on an approval was therefore
invisible in the conversation that started it, and the only way to unblock it
was to notice the sidebar badge and go to the Inbox.

The chat already shows what it handed off. This asks the same question of the
pending items: `GET /{source}/inbox` joins them against the sessions of this
conversation's running async jobs, so "whose is this" has one answer, in the
same place `/{source}/tasks` answers it for a task. The client is left with a
list to render, not a correlation to guess. The live path adds no event — the
existing `approval_requested` / `clarification_*` broadcasts already reach every
socket of the user, and re-reading the endpoint turns a nudge into something
renderable and survives a reload for free.

The card sits above the task strip rather than in the transcript: the task that
is asking may have been started twenty messages ago, and a card that scrolls
away is a card that gets missed. One at a time, with a count of what is behind
it — a blocked task stays blocked whether or not its card is on screen, so
stacking them would trade a readable chat for a queue nobody asked to see. And
it closes: the ✕ hides the card without resolving anything, leaving the item in
the Inbox, because a panel that cannot be moved takes the chat hostage.

`InboxCardsMixin` is the cards and their resolve calls, split out of
`InboxMixin` so the chat and the Inbox render the same approval rather than two
drifting copies of it; `_afterInboxResolve` is the only thing they disagree on.

Elicitations are left out: `PendingElicitationInfo` carries no `session_id`, so
there is nothing to attribute one to a task with.

Also: an async task's context label said "CronJob:", which sends whoever reads
the approval looking on the wrong page — and now says so next to the task's
real name.

docs/index.md

@@ -13,7 +13,7 @@ This index will grow over time. Right now it covers memory, projects, background
 | [memory.md](memory.md) | Private and shared memory: what goes where, the indexes and history log, why some shared facts can't be changed on request |
 | [projects.md](projects.md) | Projects: shared folders with their own assistant chat, a live file explorer, and member sharing |
 | [system-agents.md](system-agents.md) | Background agents that run on a schedule (event triage, the two memory lints, the nightly conversation review of a supervised account): what they watch, why they only ever report, why a run can be skipped, and their settings |
-| [tasks.md](tasks.md) | Background tasks: the strip above the message box, following one live, stopping one, and how every outcome comes back to the conversation |
+| [tasks.md](tasks.md) | Background tasks: the strip above the message box, following one live, stopping one, answering the approvals and questions they raise, and how every outcome comes back to the conversation |
 | [settings.md](settings.md) | The admin's Config page: interface language, the compaction model picker, debug mode |
 | [access.md](access.md) | Who can use which plugin or connector: the open default, removing access per person, and the role switch that keeps children out of it |
 | [connectors.md](connectors.md) | Connectors (MCP servers): shared vs per-user, setting one up in the UI, the sign-in and QR-pairing flows, and what to do when one is not working |