iOS: device sync (Swift port of the Android engine) behind the sync_service channel

PROPFIND walk / diff / GET / PUT mirror with conflicts, per-account state,
BGTaskScheduler background runs, and conflict notifications with Keep local /
Use server. Reviewed by Codex against the Kotlin original; its data-loss
findings are fixed and each has a regression test (fake-server end-to-end
tests, 68 Swift tests total): truncated/unusable PROPFIND answers are never a
manifest, unreadable state aborts the run, server paths can't leave the mirror,
case/Unicode collisions are skipped, atomic downloads, a locally edited file
deleted on the server is kept, '/' scope matching, serialised removal,
tracked/cancellable background runs, per-account conflicts, and sign-out vs
background-off credentials (cancel gains a 'forget' flag).

Co-Authored-By: Claude Sonnet 5.5 <noreply@anthropic.com>

ios/Runner/AppDelegate.swift

@@ -9,11 +9,25 @@ import UserNotifications
     didFinishLaunchingWithOptions launchOptions: [UIApplication.LaunchOptionsKey: Any]?
   ) -> Bool {
     TransferManager.shared.reconnect()
+    // BGTask handlers have to be registered before launch finishes.
+    SyncCoordinator.shared.registerBackgroundTasks()
+    SyncCoordinator.shared.registerNotificationCategories()
     let launched = super.application(application, didFinishLaunchingWithOptions: launchOptions)
     UNUserNotificationCenter.current().delegate = self
+    SyncCoordinator.shared.scheduleBackgroundWork()
     return launched
   }
 
+  /// Taps on a sync-conflict notification's "Keep local" / "Use server".
+  override func userNotificationCenter(
+    _ center: UNUserNotificationCenter,
+    didReceive response: UNNotificationResponse,
+    withCompletionHandler completionHandler: @escaping () -> Void
+  ) {
+    if SyncCoordinator.shared.handleNotificationResponse(response, completion: completionHandler) { return }
+    super.userNotificationCenter(center, didReceive: response, withCompletionHandler: completionHandler)
+  }
+
   /// A transfer summary ("Downloaded 1 file") usually finishes while the app
   /// is open; without this iOS delivers it silently to Notification Center
   /// instead of showing the banner.