### 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
file->set_current_path(scene->get_scene_file_path());
			};
			file->set_title(TTR("Pick a Main Scene"));
			file->popup_file_dialog();
		} break;
		case HELP_SEARCH: {
			emit_signal(SNAME("request_help_search"), "");
		} break;
		case HELP_COMMAND_PALETTE: {
			command_palette->open_popup();
		} break;
		case HELP_DOCS: {
			OS::get_singleton()->shell_open(VERSION_DOCS_URL "/");
		} break;
		case HELP_QA: {
			OS::get_singleton()->shell_open("https:godotengine.org/qa/");
		} break;
		case HELP_REPORT_A_BUG: {
			OS::get_singleton()->shell_open("https:github.com/godotengine/godot/issues");
		} break;
		case HELP_COPY_SYSTEM_INFO: {
			String info = _get_system_info();
			DisplayServer::get_singleton()->clipboard_set(info);
		} break;
		case HELP_SUGGEST_A_FEATURE: {
			OS::get_singleton()->shell_open("https:github.com/godotengine/godot-proposals#readme");
		} break;
		case HELP_SEND_DOCS_FEEDBACK: {
			OS::get_singleton()->shell_open("https:github.com/godotengine/godot-docs/issues");
		} break;