Summary

• keep session.error and session.deleted tasks in memory until the parent notification queue runs, so terminal tasks do not disappear before their result can be surfaced
• move pending-parent cleanup to notification time so simultaneous completions only emit one final all-complete notification
• skip terminal tasks during TTL pruning and cover the regressions with background-agent tests

Testing

• bun test src/features/background-agent/
• bun run typecheck
• bun run build

Closes #2292

Summary by cubic

Fixes #2292 by keeping terminal and stale-pruned background tasks in memory until the parent notification runs, so results don’t disappear. Cleanup is deferred to notification time; TTL pruning skips terminal tasks and routes stale running tasks through a final notification.

• Bug Fixes
  • Keep completed/error/cancelled/interrupt tasks until notifyParentSession runs.
  • Defer pendingByParent cleanup to notification; coalesce into one final “all complete” and clean even if promptAsync fails.
  • TTL pruning: skip terminal tasks; mark stale running tasks as error, notify parent, and delay cleanup.
  • On session.deleted: cancel descendants, remove toast entries, keep tasks and timers until delayed cleanup, and drop pending notifications for the deleted parent.

^{Written for commit d46946c. Summary will update on new commits.}