### 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)

config["fileSizes"] = p_file_sizes;
	String head_include;
	if (p_preset->get("html/export_icon")) {
		head_include += "<link id='-gd-engine-icon' rel='icon' type='image/png' href='" + p_name + ".icon.png' />\n";
		head_include += "<link rel='apple-touch-icon' href='" + p_name + ".apple-touch-icon.png'/>\n";
	}
	if (p_preset->get("progressive_web_app/enabled")) {
		head_include += "<link rel='manifest' href='" + p_name + ".manifest.json'>\n";
		config["serviceWorker"] = p_name + ".service.worker.js";
	}
	
	const String str_config = Variant(config).to_json_string();
	const String custom_head_include = p_preset->get("html/head_include");
	HashMap<String, String> replaces;
	replaces["$GODOT_URL"] = p_name + ".js";
	replaces["$GODOT_PROJECT_NAME"] = GLOBAL_GET("application/config/name");
	replaces["$GODOT_HEAD_INCLUDE"] = head_include + custom_head_include;
	replaces["$GODOT_CONFIG"] = str_config;
	_replace_strings(replaces, p_html);
}
Error EditorExportPlatformWeb::_add_manifest_icon(const String &p_path, const String &p_icon, int p_size, Array &r_arr) {
	const String name = p_path.get_file().get_basename();
	const String icon_name = vformat("%s.%dx%d.png", name, p_size, p_size);
	const String icon_dest = p_path.get_base_dir().path_join(icon_name);
	Ref<Image> icon;
	if (!p_icon.is_empty()) {
		icon.instantiate();
		const Error err = ImageLoader::load_image(p_icon, icon);
		if (err != OK) {
			add_message(EXPORT_MESSAGE_ERROR, TTR("Icon Creation"), vformat(TTR("Could not read file: \"%s\"."), p_icon));