Describe the problem

There's currently no way to tell if there are new pinned messages you haven't seen. We should add an unread badge to the pin button.

Describe the solution you'd like

Store a small room account data event with:

{
    "hash": "a3f9b2c1d4",
    "count": 7,
    "last_seen_id": "$abc123"
}

hash is the first 10 characters of the SHA-256 of the sorted pin ID list. Compared against the current pin list on sync to detect if anything changed without fetching individual events. Hash detects changes, count shows "+N new", and last_seen_id highlights new pins inside the panel (since pin order is guaranteed by spec). If last_seen_id was unpinned, fall back to hash comparison.

Alternatives considered

Stopgap until MSC4238 is merged.

Additional context

No response