feat: keep background Updates badge current (Closes #1281)

src/push_notifications.py

@@ -133,6 +133,7 @@ async def dispatch_unread_updates(
             for item in page.get("items", [])
             if isinstance(item, dict) and str(item.get("id", "")).isdigit()
         }
+        unread_count = min(len(thread_revisions), 9999)
         await asyncio.to_thread(store.reconcile_unread, thread_revisions)
         deliveries = await asyncio.to_thread(store.claim_unseen, thread_revisions)
         if session_statuses is not None:
@@ -201,6 +202,7 @@ async def dispatch_unread_updates(
                             "route": f"#/my-work/update/{thread_id}",
                             "tag": f"stackchain-update-{thread_id}",
                             "notification_id": thread_id,
+                            "unread_count": unread_count,
                         },
                         separators=(",", ":"),
                     )
@@ -258,6 +260,7 @@ async def dispatch_unread_updates(
                             "route": "#/my-work/updates",
                             "tag": "stackchain-update-digest",
                             "update_count": len(overflow_revisions),
+                            "unread_count": unread_count,
                         },
                         separators=(",", ":"),
                     )