feat(desktop): unread-chat count badge on the taskbar icon

Mirrors the rail's unread-chat count onto the Windows taskbar icon (overlay) and the
macOS/Linux dock badge. updateRailUnread() draws a small red count badge on a canvas
and hands it to the shell via bizConnectNative.setUnread(count, dataUrl); main sets it
with win.setOverlayIcon + app.setBadgeCount. Clears at 0, shows 99+ past 99. No-op in a
browser/mobile. (home.html build batch17)

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

desktop/main.js

@@ -8,7 +8,7 @@
 //   - external links open in the user's browser, not inside the app
 //
 // Server origin is configurable so the same build works against prod or a dev server.
-const { app, BrowserWindow, session, desktopCapturer, shell, Menu, ipcMain } = require('electron');
+const { app, BrowserWindow, session, desktopCapturer, shell, Menu, ipcMain, nativeImage } = require('electron');
 const path = require('path');
 
 // Renderer asks (via preload) to raise the window — e.g. when an OS notification is clicked.
@@ -19,6 +19,18 @@ ipcMain.on('focus-window', () => {
   win.focus();
 });
 
+// Unread badge on the taskbar icon. The renderer computes the count (chats with unread) and
+// draws the badge image (it has a canvas); Windows shows it via an overlay icon, macOS/Linux
+// via the dock badge count.
+ipcMain.on('set-unread', (_e, { count, dataUrl } = {}) => {
+  try {
+    if (typeof app.setBadgeCount === 'function') app.setBadgeCount(count || 0); // macOS/Linux dock
+    if (!win) return;
+    const overlay = (count > 0 && dataUrl) ? nativeImage.createFromDataURL(dataUrl) : null;
+    win.setOverlayIcon(overlay, count > 0 ? (count + ' unread chats') : ''); // Windows taskbar
+  } catch (_) {}
+});
+
 const SERVER_URL = (process.env.SERVER_URL || 'https://remote.bizgaze.com').replace(/\/+$/, '');
 
 let win;