### 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
}
			_proceed_closing_scene_tabs();
		} break;
		case FILE_QUIT: {
			project_run_bar->stop_playing();
			_exit_editor(EXIT_SUCCESS);
		} break;
		case RUN_PROJECT_MANAGER: {
			project_run_bar->stop_playing();
			_exit_editor(EXIT_SUCCESS);
			String exec = OS::get_singleton()->get_executable_path();
			List<String> args;
			for (const String &a : Main::get_forwardable_cli_arguments(Main::CLI_SCOPE_TOOL)) {
				args.push_back(a);
			}
			String exec_base_dir = exec.get_base_dir();
			if (!exec_base_dir.is_empty()) {
				args.push_back("--path");
				args.push_back(exec_base_dir);
			}
			args.push_back("--project-manager");
			Error err = OS::get_singleton()->create_instance(args);
			ERR_FAIL_COND(err);
		} break;
		case RELOAD_CURRENT_PROJECT: {
			restart_editor();
		} break;
	}
}
void EditorNode::_update_file_menu_opened() {