feat: desktop tray unread overlay + taskbar flash on new mention

Extends useTauriNotificationBadge to also invoke set_tray_unread (mirror the
unread-mention state onto the tray icon, visible when minimized) and flash_window
(flash the taskbar button when new mentions arrive while the window is unfocused).
No-op outside Tauri.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

src/app/hooks/useTauriNotificationBadge.ts

@@ -1,4 +1,4 @@
-import { useEffect } from 'react';
+import { useEffect, useRef } from 'react';
 import { useAtomValue } from 'jotai';
 import { roomToUnreadAtom } from '../state/room/roomToUnread';
 
@@ -10,6 +10,7 @@ const tauriInvoke = (): TauriInternals['invoke'] | undefined =>
 
 export function useTauriNotificationBadge() {
   const roomToUnread = useAtomValue(roomToUnreadAtom);
+  const prevHighlightsRef = useRef(0);
 
   useEffect(() => {
     const invoke = tauriInvoke();
@@ -21,5 +22,14 @@ export function useTauriNotificationBadge() {
     });
 
     invoke('set_badge_count', { count: totalHighlights }).catch(() => {});
+    // Mirror the unread state onto the tray icon (visible when minimized to tray).
+    invoke('set_tray_unread', { unread: totalHighlights > 0 }).catch(() => {});
+
+    // Flash the taskbar button when new mentions arrive and the window is not
+    // focused, then reset the baseline.
+    if (totalHighlights > prevHighlightsRef.current && !document.hasFocus()) {
+      invoke('flash_window').catch(() => {});
+    }
+    prevHighlightsRef.current = totalHighlights;
   }, [roomToUnread]);
 }