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.

ViewModels/MainViewModel.cs

@@ -616,6 +616,25 @@ public partial class MainViewModel : ViewModelBase
     }
 
 
+    /// <summary>
+    /// Ends a running session before the app tears itself down, and says whether it is safe to close.
+    ///
+    /// <para>Closing with a recording active used to dispose the frame pump and the audio mixer under a
+    /// live loop: ffmpeg never flushed, the file was never renamed or kept, and the pump was disposed
+    /// while it was still working. The window now holds itself open until the ordinary stop completes
+    /// (bounded — a stuck encoder must not trap the creator in an unclosable app).</para>
+    /// </summary>
+    internal async Task<bool> StopBeforeShutdownAsync()
+    {
+        if (!IsLive && !IsRecording) return true;
+
+        StopStream();
+        for (var i = 0; i < 120 && (IsLive || IsRecording); i++)   // up to ~6 s
+            await Task.Delay(50).ConfigureAwait(true);
+
+        return !IsLive && !IsRecording;
+    }
+
     public void Shutdown()
     {
         StopCurrentLaunchLogFeed();