Notifications: add badge dot to the system tray icon when there is a new or unread message.

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Is your feature request related to a problem? Please describe.

There is a notification about a new message and taskbar icon badge. But when the window is closed and the notification is closed, it is not visible that some notification is missed.

Describe the solution you'd like

Add a badge dot to the tray system tray icon when there is a new or unread message.

Describe alternatives you've considered

Currently, we have a badge about a new notification similar to * in the title on web-client. But it could be more useful to have a badge about unread mentions.

The difference is that the current badge appears only on a notification and disappears on window/tab activating. While usually such badges are shown when there are any unread mentions, no matter if the window is open or it is an old message.

Additional context

Discord example:

Start by tracing the desktop client's system-tray icon implementation and the existing notification or unread-message state. Done means the tray icon shows a badge dot whenever a new or unread message exists, including after the window and notification are closed, and clears according to the intended unread behavior.