### Tested versions

v4.2.stable.official [46dc27791]

### System information

Windows 11- Godot v4.2.stable.official [46dc27791] - Vulkan (Forward+)

### Issue description

When I switch picker mode the next time I open it should be the pick mode I selected last time but it's not

https://github.com/godotengine/godot/assets/46812028/522c8b55-14cf-4781-963f-a465a03c9432



### Steps to reproduce

Toggle a picker mode off and on again

### Minimal reproduction project (MRP)

N/A
Array rc = EditorSettings::get_singleton()->get_project_metadata("recent_files", "scenes", Array());
	recent_scenes->clear();
	String path;
	for (int i = 0; i < rc.size(); i++) {
		path = rc[i];
		recent_scenes->add_item(path.replace("res:", ""), i);
	}
	recent_scenes->add_separator();
	recent_scenes->add_shortcut(ED_SHORTCUT("editor/clear_recent", TTR("Clear Recent Scenes")));
	recent_scenes->reset_size();
}
void EditorNode::_quick_opened() {
	Vector<String> files = quick_open->get_selected_files();
	bool open_scene_dialog = quick_open->get_base_type() == "PackedScene";
	for (int i = 0; i < files.size(); i++) {
		String res_path = files[i];
		if (open_scene_dialog || ClassDB::is_parent_class(ResourceLoader::get_resource_type(res_path), "PackedScene")) {
			open_request(res_path);
		} else {
			load_resource(res_path);
		}
	}
}
void EditorNode::_project_run_started() {