### 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 err;
		}
		if (icon->get_width() != p_size || icon->get_height() != p_size) {
			icon->resize(p_size, p_size);
		}
	} else {
		icon = _get_project_icon();
		icon->resize(p_size, p_size);
	}
	const Error err = icon->save_png(icon_dest);
	if (err != OK) {
		add_message(EXPORT_MESSAGE_ERROR, TTR("Icon Creation"), vformat(TTR("Could not write file: \"%s\"."), icon_dest));
		return err;
	}
	Dictionary icon_dict;
	icon_dict["sizes"] = vformat("%dx%d", p_size, p_size);
	icon_dict["type"] = "image/png";
	icon_dict["src"] = icon_name;
	r_arr.push_back(icon_dict);
	return err;
}
Error EditorExportPlatformWeb::_build_pwa(const Ref<EditorExportPreset> &p_preset, const String p_path, const Vector<SharedObject> &p_shared_objects) {
	String proj_name = GLOBAL_GET("application/config/name");
	if (proj_name.is_empty()) {
		proj_name = "Godot Game";
	}
	
	const String dir = p_path.get_base_dir();
	const String name = p_path.get_file().get_basename();
	bool extensions = (bool)p_preset->get("variant/extensions_support");