### 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)

HashMap<String, String> replaces;
	replaces["@GODOT_VERSION@"] = String::num_int64(OS::get_singleton()->get_unix_time()) + "|" + String::num_int64(OS::get_singleton()->get_ticks_usec());
	replaces["@GODOT_NAME@"] = proj_name.substr(0, 16);
	replaces["@GODOT_OFFLINE_PAGE@"] = name + ".offline.html";
	
	Array cache_files;
	cache_files.push_back(name + ".html");
	cache_files.push_back(name + ".js");
	cache_files.push_back(name + ".offline.html");
	if (p_preset->get("html/export_icon")) {
		cache_files.push_back(name + ".icon.png");
		cache_files.push_back(name + ".apple-touch-icon.png");
	}
	cache_files.push_back(name + ".worker.js");
	cache_files.push_back(name + ".audio.worklet.js");
	replaces["@GODOT_CACHE@"] = Variant(cache_files).to_json_string();
	
	Array opt_cache_files;
	opt_cache_files.push_back(name + ".wasm");
	opt_cache_files.push_back(name + ".pck");
	if (extensions) {
		opt_cache_files.push_back(name + ".side.wasm");
		for (int i = 0; i < p_shared_objects.size(); i++) {
			opt_cache_files.push_back(p_shared_objects[i].path.get_file());
		}
	}
	replaces["@GODOT_OPT_CACHE@"] = Variant(opt_cache_files).to_json_string();
	const String sw_path = dir.path_join(name + ".service.worker.js");
	Vector<uint8_t> sw;
	{