fix(native): Wave-2 audit fixes (D1, D3, D5)

- D1 (HIGH): Linux no-sleep was completely non-functional — the zbus
  ScreenSaver inhibit was bound to a function-local D-Bus connection dropped on
  return, so the screensaver service auto-released it instantly. Keep a
  long-lived Connection in managed state (InhibitState { conn, cookie }) so the
  same connection holds Inhibit and issues UnInhibit; created once, reused.
- D3: tray "Do Not Disturb" desynced from the web manualDndAtom after any reload
  (custom-chrome toggle / logout) — the atom is in-memory and reset while the
  tray stayed checked. Added TrayDndState + a get_tray_dnd command so the web
  hook re-hydrates the atom on mount.
- D5: install_update now calls app.restart() after a successful install so the
  new version actually runs (Linux AppImage kept running the old binary; the UI
  hung on "installing").

CI-compile-verified (windows + linux). Web-side wiring (get_tray_dnd query,
updater terminal state) landed on cinny:lotus.

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

src-tauri/src/lib.rs

@@ -138,6 +138,13 @@ async fn install_update(app: tauri::AppHandle) -> Result<(), String> {
                 .download_and_install(|_chunk, _total| {}, || {})
                 .await
                 .map_err(|e| e.to_string())?;
+            // Only reached on a successful download+install (the `?` above bails
+            // otherwise). Relaunch so the freshly installed version actually
+            // runs — without this the UI hangs on "installing", and on a Linux
+            // AppImage the running process is still the old binary. `restart()`
+            // exits the current process and never returns, so nothing after it
+            // runs for the update case.
+            app.restart();
         }
     }
     Ok(())
@@ -368,6 +375,23 @@ struct TrayUnreadState {
     height: u32,
 }
 
+/// Holds a clone of the tray "Do Not Disturb" `CheckMenuItem` so `get_tray_dnd`
+/// can read its live checkstate. The tray only emits `lotus-dnd-changed` on
+/// click, but the web `manualDndAtom` is in-memory and resets on every reload,
+/// so the web hook re-hydrates from this on mount. `CheckMenuItem` is a cheap
+/// clonable handle to the same underlying menu item.
+struct TrayDndState(CheckMenuItem<tauri::Wry>);
+
+/// Return the tray DND toggle's current checkstate so the web side can
+/// re-hydrate `manualDndAtom` after a reload. Returns `false` when the tray
+/// wasn't created (e.g. missing bundled icon) rather than erroring the call.
+#[tauri::command]
+fn get_tray_dnd(app: tauri::AppHandle) -> bool {
+    app.try_state::<TrayDndState>()
+        .map(|s| s.0.is_checked().unwrap_or(false))
+        .unwrap_or(false)
+}
+
 /// Paint a small white-ringed red "unread" dot into the bottom-right corner of
 /// an RGBA buffer, in place. Cross-platform (operates on raw pixels).
 fn draw_unread_dot(rgba: &mut [u8], width: u32, height: u32) {
@@ -448,6 +472,7 @@ pub fn run() {
         .invoke_handler(tauri::generate_handler![
             set_badge_count,
             set_tray_unread,
+            get_tray_dnd,
             flash_window,
             send_notification,
             check_for_update,
@@ -563,6 +588,10 @@ pub fn run() {
                     width,
                     height,
                 });
+
+                // Keep a handle to the DND CheckMenuItem so `get_tray_dnd` can
+                // report its live checkstate for web re-hydration after reload.
+                app.manage(TrayDndState(dnd_item));
             } else {
                 eprintln!("tray: no bundled window icon; skipping system tray setup");
             }