### 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)

const String tls_cert = EDITOR_GET("export/web/tls_certificate");
	
	{
		MutexLock lock(server_lock);
		server->stop();
		err = server->listen(bind_port, bind_ip, use_tls, tls_key, tls_cert);
	}
	if (err != OK) {
		add_message(EXPORT_MESSAGE_ERROR, TTR("Run"), vformat(TTR("Error starting HTTP server: %d."), err));
		return err;
	}
	OS::get_singleton()->shell_open(String((use_tls ? "https://" : "http://") + bind_host + ":" + itos(bind_port) + "/tmp_js_export.html"));
	
	
	return OK;
}
Ref<Texture2D> EditorExportPlatformWeb::get_run_icon() const {
	return run_icon;
}
void EditorExportPlatformWeb::_server_thread_poll(void *data) {
	EditorExportPlatformWeb *ej = static_cast<EditorExportPlatformWeb *>(data);
	while (!ej->server_quit) {
		OS::get_singleton()->delay_usec(6900);
		{
			MutexLock lock(ej->server_lock);
			ej->server->poll();
		}
	}
}
EditorExportPlatformWeb::EditorExportPlatformWeb() {