### 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
} break;
		case SETTINGS_INSTALL_ANDROID_BUILD_TEMPLATE: {
			gradle_build_manage_templates->hide();
			file_android_build_source->popup_centered_ratio();
		} break;
		case SETTINGS_MANAGE_FEATURE_PROFILES: {
			feature_profile_manager->popup_centered_clamped(Size2(900, 800) * EDSCALE, 0.8);
		} break;
		case SETTINGS_TOGGLE_FULLSCREEN: {
			DisplayServer::WindowMode mode = DisplayServer::get_singleton()->window_get_mode();
			if (mode == DisplayServer::WINDOW_MODE_FULLSCREEN || mode == DisplayServer::WINDOW_MODE_EXCLUSIVE_FULLSCREEN) {
				DisplayServer::get_singleton()->window_set_mode(prev_mode);
			} else {
				prev_mode = mode;
				DisplayServer::get_singleton()->window_set_mode(DisplayServer::WINDOW_MODE_FULLSCREEN);
			}
		} break;
		case EDITOR_SCREENSHOT: {
			screenshot_timer->start();
		} break;
		case SETTINGS_PICK_MAIN_SCENE: {
			file->set_file_mode(EditorFileDialog::FILE_MODE_OPEN_FILE);
			List<String> extensions;
			ResourceLoader::get_recognized_extensions_for_type("PackedScene", &extensions);
			file->clear_filters();
			for (int i = 0; i < extensions.size(); i++) {
				file->add_filter("*." + extensions[i], extensions[i].to_upper());
			}
			Node *scene = editor_data.get_edited_scene_root();
			if (scene) {