### Tested versions

Godot Engine v4.2.stable.official.46dc27791

### System information

Windows 10

### Issue description

When exporting to web, Godot fails with this error in the console:

> scene/main/window.cpp:884 - Attempting to make child window exclusive, but the parent window already has another exclusive child. This window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559/@<!-- -->VBoxContainer@<!-- -->137/@<!-- -->AcceptDialog@<!-- -->411, parent window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559, current exclusive child window: /root/@<!-- -->EditorNode@<!-- -->17120/@<!-- -->Panel@<!-- -->13/@<!-- -->ProjectExportDialog@<!-- -->559/@<!-- -->ProgressDialog@<!-- -->17444

![a](https://github.com/godotengine/godot/assets/2000051/bf9d22aa-d3fe-4e11-bff6-c387e740fe90)

This happens for newly created empty projects.


### Steps to reproduce

1. Open the attached project (or create a new project).
2. Project->Export->Export All
3. Observe the console error.  No output files are generated.


### Minimal reproduction project (MRP)

[export_test.zip](https://github.com/godotengine/godot/files/13574810/export_test.zip)

return p_index == 1 ? stop_icon : EditorExportPlatform::get_option_icon(p_index);
}
int EditorExportPlatformWeb::get_options_count() const {
	return menu_options;
}
Error EditorExportPlatformWeb::run(const Ref<EditorExportPreset> &p_preset, int p_option, int p_debug_flags) {
	if (p_option == 1) {
		MutexLock lock(server_lock);
		server->stop();
		return OK;
	}
	const String dest = EditorPaths::get_singleton()->get_cache_dir().path_join("web");
	Ref<DirAccess> da = DirAccess::create(DirAccess::ACCESS_FILESYSTEM);
	if (!da->dir_exists(dest)) {
		Error err = da->make_dir_recursive(dest);
		if (err != OK) {
			add_message(EXPORT_MESSAGE_ERROR, TTR("Run"), vformat(TTR("Could not create HTTP server directory: %s."), dest));
			return err;
		}
	}