### 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)

Dictionary file_sizes;
	Ref<FileAccess> f = FileAccess::open(pck_path, FileAccess::READ);
	if (f.is_valid()) {
		file_sizes[pck_path.get_file()] = (uint64_t)f->get_length();
	}
	f = FileAccess::open(base_path + ".wasm", FileAccess::READ);
	if (f.is_valid()) {
		file_sizes[base_name + ".wasm"] = (uint64_t)f->get_length();
	}
	
	const String html_path = custom_html.is_empty() ? base_path + ".html" : custom_html;
	Vector<uint8_t> html;
	f = FileAccess::open(html_path, FileAccess::READ);
	if (f.is_null()) {
		add_message(EXPORT_MESSAGE_ERROR, TTR("Export"), vformat(TTR("Could not read HTML shell: \"%s\"."), html_path));
		return ERR_FILE_CANT_READ;
	}
	html.resize(f->get_length());
	f->get_buffer(html.ptrw(), html.size());
	f.unref(); 
	
	_fix_html(html, p_preset, base_name, p_debug, p_flags, shared_objects, file_sizes);
	Error err = _write_or_error(html.ptr(), html.size(), p_path);
	if (err != OK) {
		return err;
	}
	html.resize(0);
	
	Ref<Image> splash = _get_project_splash();
	const String splash_png_path = base_path + ".png";