### 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)

if (splash->save_png(splash_png_path) != OK) {
		add_message(EXPORT_MESSAGE_ERROR, TTR("Export"), vformat(TTR("Could not write file: \"%s\"."), splash_png_path));
		return ERR_FILE_CANT_WRITE;
	}
	
	
	if (export_icon) {
		Ref<Image> favicon = _get_project_icon();
		const String favicon_png_path = base_path + ".icon.png";
		if (favicon->save_png(favicon_png_path) != OK) {
			add_message(EXPORT_MESSAGE_ERROR, TTR("Export"), vformat(TTR("Could not write file: \"%s\"."), favicon_png_path));
			return ERR_FILE_CANT_WRITE;
		}
		favicon->resize(180, 180);
		const String apple_icon_png_path = base_path + ".apple-touch-icon.png";
		if (favicon->save_png(apple_icon_png_path) != OK) {
			add_message(EXPORT_MESSAGE_ERROR, TTR("Export"), vformat(TTR("Could not write file: \"%s\"."), apple_icon_png_path));
			return ERR_FILE_CANT_WRITE;
		}
	}
	
	if (pwa) {
		err = _build_pwa(p_preset, p_path, shared_objects);
		if (err != OK) {
			return err;
		}
	}
	return OK;
}
bool EditorExportPlatformWeb::poll_export() {