### 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)

Ref<EditorExportPreset> current = get_current_preset();
	if (current.is_null()) {
		return;
	}
	TreeItem *item = include_files->get_edited();
	String path = item->get_metadata(0);
	EditorExportPreset::FileExportMode file_export_mode = (EditorExportPreset::FileExportMode)p_id;
	current->set_file_export_mode(path, file_export_mode);
	item->set_metadata(1, file_export_mode);
	_propagate_file_export_mode(include_files->get_root(), EditorExportPreset::MODE_FILE_NOT_CUSTOMIZED);
}
void ProjectExportDialog::_export_pck_zip() {
	Ref<EditorExportPreset> current = get_current_preset();
	ERR_FAIL_COND(current.is_null());
	String dir = current->get_export_path().get_base_dir();
	export_pck_zip->set_current_dir(dir);
	export_pck_zip->popup_file_dialog();
}
void ProjectExportDialog::_export_pck_zip_selected(const String &p_path) {
	Ref<EditorExportPreset> current = get_current_preset();
	ERR_FAIL_COND(current.is_null());
	Ref<EditorExportPlatform> platform = current->get_platform();
	ERR_FAIL_COND(platform.is_null());
	if (p_path.ends_with(".zip")) {
		platform->export_zip(current, export_pck_zip_debug->is_pressed(), p_path);
	} else if (p_path.ends_with(".pck")) {
		platform->export_pack(current, export_pck_zip_debug->is_pressed(), p_path);
	}
}
void ProjectExportDialog::_open_export_template_manager() {