[FEATURE] Claude Desktop (Linux): show an unread/attention count on the tray icon

Problem

When several sessions are running, desktop notifications become very frequent. Each one pops up over whatever I'm reading, which hurts usability — but turning notifications off today means losing any signal that a session finished or needs my input.

Proposal

Show a counter next to (or overlaid on) the tray icon with the number of pending notifications — sessions that finished or are waiting for input and haven't been looked at yet. This is the common pattern in messaging apps (Teams, WhatsApp, Slack, Telegram): the tray icon carries the unread count, and it clears as the items are viewed.

With that in place I could disable the popup notifications entirely and rely on a glance at the tray, without being interrupted.

Notes

• The app already tracks an unread count for the dock/taskbar badge (see #87700), so the number exists; on Linux it just isn't surfaced anywhere visible when the window is closed to the tray.
• On Linux this can be done by swapping the tray image for one with the count drawn on it (tray.setImage), which works with StatusNotifierItem hosts that don't support labels. A tooltip such as "Claude — 3 sessions need attention" would complement it.
• Ideally independent of the notification setting, so the counter keeps working with popups turned off.

Environment

• Claude Desktop 2.9939.4 (official claude-desktop .deb)
• Linux Mint 22.3, Cinnamon 6.6.9, X11

Related: #98678 (left click on the tray icon doesn't show the window).