### 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
install_android_build_template->popup_centered();
				} else {
					gradle_build_manage_templates->popup_centered();
				}
			}
		} break;
		case TOOLS_BUILD_PROFILE_MANAGER: {
			build_profile_manager->popup_centered_clamped(Size2(700, 800) * EDSCALE, 0.8);
		} break;
		case RUN_USER_DATA_FOLDER: {
			
			OS::get_singleton()->ensure_user_data_dir();
			OS::get_singleton()->shell_show_in_file_manager(OS::get_singleton()->get_user_data_dir(), true);
		} break;
		case FILE_EXPLORE_ANDROID_BUILD_TEMPLATES: {
			OS::get_singleton()->shell_show_in_file_manager(ProjectSettings::get_singleton()->get_resource_path().path_join("android"), true);
		} break;
		case FILE_QUIT:
		case RUN_PROJECT_MANAGER:
		case RELOAD_CURRENT_PROJECT: {
			if (p_confirmed && plugin_to_save) {
				plugin_to_save->save_external_data();
				p_confirmed = false;
			}
			if (!p_confirmed) {
				bool save_each = EDITOR_GET("interface/editor/save_each_scene_on_quit");
				if (_next_unsaved_scene(!save_each) == -1) {
					if (EditorUndoRedoManager::get_singleton()->is_history_unsaved(EditorUndoRedoManager::GLOBAL_HISTORY)) {
						if (p_option == RELOAD_CURRENT_PROJECT) {
							save_confirmation->set_ok_button_text(TTR("Save & Reload"));