[desktop_multi_window] Changes to widgets in sub-window not reflected after hot reload/restart

When you make a change to a widget in the main window and press save, it is immediately reflected (using Visual Studio Code and Flutter extension). For widgets in the sub window, the changes aren't seen until completely stopping and restarting the debug session. Closing and opening the sub window also does not even show the change

Reproduce Steps

Steps to reproduce the behavior:

1. Create new Flutter application
2. Implement multi_window following instructions / example
3. Start the app using VSCode debugger
4. Change any widget in the sub window (example: modify App Bar title)
5. Press save
6. See that change was not reflected

Expected behavior

Should see the change made be reflected in the sub window. Note that this works if you download and run the "example" directory, but starting a brand new project doesn't.

This example project I created demonstrates the issue. Note that also saving seems to break communication (i.e. entering text on main window no longer updates until you close and reopen the window).

Version (please complete the following information):

• Flutter Version: 3.16.9
• OS: MacOS
• plugin: desktop_multi_window: 0.2.0

Start with the linked example project and the plugin's example directory, then reproduce the behavior in a newly created Flutter app using the listed steps. Compare how sub-window changes and communication behave after saving; done means hot reload or reopening the sub-window reflects edits without breaking communication.