### 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)

String md5 = FileAccess::get_md5(p_path);
	if (FileAccess::exists(p_path + ".import")) {
		
		md5 += FileAccess::get_md5(p_path + ".import");
	}
	fec.source_md5 = md5;
	
	String type = ResourceLoader::get_resource_type(p_path);
	bool modified = false;
	String save_path;
	if (type == "PackedScene") { 
		Ref<PackedScene> ps = ResourceLoader::load(p_path, "PackedScene", ResourceFormatLoader::CACHE_MODE_IGNORE);
		ERR_FAIL_COND_V(ps.is_null(), p_path);
		Node *node = ps->instantiate(PackedScene::GEN_EDIT_STATE_INSTANCE); 
		ERR_FAIL_NULL_V(node, p_path);
		if (!customize_scenes_plugins.is_empty()) {
			for (Ref<EditorExportPlugin> &plugin : customize_scenes_plugins) {
				Node *customized = plugin->_customize_scene(node, p_path);
				if (customized != nullptr) {
					node = customized;
					modified = true;
				}
			}
		}
		if (!customize_resources_plugins.is_empty()) {
			if (_export_customize_scene_resources(node, node, customize_resources_plugins)) {
				modified = true;
			}
		}
		if (modified || p_force_save) {