test(tui): cover background sync progress workflows

- split background-sync TUI regression coverage out from the feature implementation commit
- add manual-sync state-machine coverage for duplicate requests, busy workers, failures, disconnects, and completion refresh
- add non-blocking mailbox-open, sync palette dismissal, and footer progress rendering coverage for startup, manual, and auto-sync states

Signed-off-by: Chao Liu <chao.liu.zevorn@gmail.com>

Author: zevorn