### 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)

icon_path = ProjectSettings::get_singleton()->globalize_path(icon_path);
	if (p_console_icon) {
		String console_icon_path = ProjectSettings::get_singleton()->globalize_path(p_preset->get("application/console_wrapper_icon"));
		if (!console_icon_path.is_empty() && FileAccess::exists(console_icon_path)) {
			icon_path = console_icon_path;
		}
	}
	String tmp_icon_path = EditorPaths::get_singleton()->get_cache_dir().path_join("_rcedit.ico");
	if (!icon_path.is_empty()) {
		if (_process_icon(p_preset, icon_path, tmp_icon_path) != OK) {
			add_message(EXPORT_MESSAGE_WARNING, TTR("Resources Modification"), vformat(TTR("Invalid icon file \"%s\"."), icon_path));
			icon_path = String();
		}
	}
	String file_verion = p_preset->get_version("application/file_version", true);
	String product_version = p_preset->get_version("application/product_version", true);
	String company_name = p_preset->get("application/company_name");
	String product_name = p_preset->get("application/product_name");
	String file_description = p_preset->get("application/file_description");
	String copyright = p_preset->get("application/copyright");
	String trademarks = p_preset->get("application/trademarks");
	String comments = p_preset->get("application/comments");
	List<String> args;
	args.push_back(p_path);
	if (!icon_path.is_empty()) {
		args.push_back("--set-icon");
		args.push_back(tmp_icon_path);
	}
	if (!file_verion.is_empty()) {
		args.push_back("--set-file-version");