### 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
if (p_custom_action_name == "select_current") {
		Node *scene = editor_data.get_edited_scene_root();
		if (!scene) {
			show_accept(TTR("There is no defined scene to run."), TTR("OK"));
			return;
		}
		pick_main_scene->hide();
		if (!FileAccess::exists(scene->get_scene_file_path())) {
			current_menu_option = FILE_SAVE_AND_RUN_MAIN_SCENE;
			_menu_option_confirm(FILE_SAVE_AS_SCENE, true);
			file->set_title(TTR("Save scene before running..."));
		} else {
			current_menu_option = SETTINGS_PICK_MAIN_SCENE;
			_dialog_action(scene->get_scene_file_path());
		}
	}
}
Ref<Texture2D> EditorNode::_get_class_or_script_icon(const String &p_class, const Ref<Script> &p_script, const String &p_fallback, bool p_fallback_script_to_theme) {