fix(lifecycle): end the session before disposing it on close

Closing with a recording active called Shutdown() straight away, which disposed the frame pump and the
audio mixer underneath a running loop: ffmpeg never flushed, the file was never finalized or renamed,
and the pump was torn down mid-iteration. That is the 'app goes autistic on close' report.

MainWindow_Closing now holds the window open, runs StopBeforeShutdownAsync (the ordinary stop, bounded at
~6 s so a stuck encoder cannot make the app unclosable), then closes for real. StopStream already honours
the audio-before-video ordering, so the close path inherits it instead of inventing its own teardown.

Test: with nothing running, closing must not hold the window at all (the fast path stays fast).

Scope justification: Assets/starting-background.jpg is outside this unit — the creator's replacement art,
still awaiting their visual check.

MainWindow.xaml.cs

@@ -147,8 +147,28 @@ public partial class MainWindow : Window
             _topBar.RefreshAvatar();
     }
 
-    private void MainWindow_Closing(object? sender, CancelEventArgs e)
+    private bool _shutdownComplete;
+
+    private async void MainWindow_Closing(object? sender, CancelEventArgs e)
     {
+        if (_shutdownComplete)
+        {
+            SaveWindowGeometry();
+            _viewModel.Shutdown();
+            return;
+        }
+
+        // A session must END before anything is disposed (2026-11-12: closing mid-recording used to
+        // pull the pump out from under itself). Hold the window open, stop properly, then close again.
+        if (_viewModel.IsLive || _viewModel.IsRecording)
+        {
+            e.Cancel = true;
+            await _viewModel.StopBeforeShutdownAsync();
+            _shutdownComplete = true;
+            Close();
+            return;
+        }
+
         SaveWindowGeometry();
         _viewModel.Shutdown();
     }