Bug: Notification badge count persists after all notifications cleared (Windows)

Description

The Copilot desktop app (Windows) shows a notification badge count on the system tray/taskbar icon even when there are no outstanding notifications, pending plan approvals, or inbox entries.

Steps to Reproduce

1. Use the Copilot desktop app normally
2. Clear/read all notifications and pending items
3. Observe the app icon still shows a numeric badge (e.g. '1' or '2')

Expected Behavior

The badge count should be 0 (no badge shown) when there are no unread notifications or pending items.

Actual Behavior

The badge persists with a stale count. It does not decrement to zero even after:

• Viewing all notifications
• Confirming inbox is empty (0 entries in session database)
• Reinstalling the app
• Clearing all caches
• App self-updating the database

Environment

• OS: Windows
• Copilot App Version: 1.0.78-2
• The issue persists across reinstalls and cache clears

Additional Context

This appears to be a state management issue where the badge count is not properly synced with the actual notification state.

ファイル、テスト、エントリーポイントは特定されていません。Windows で古い badge を再現し、通知と保留中の項目の状態がトレイまたはタスクバーのカウントをどのように更新するかを追跡してください。通知と保留中の項目をクリアしたときに badge がゼロになれば完了です。再インストール後やキャッシュクリアのシナリオも含みます。