### 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_UPDATE_CONTINUOUSLY: {
			EditorSettings::get_singleton()->set("interface/editor/update_continuously", true);
			_update_update_spinner();
			show_accept(TTR("This option is deprecated. Situations where refresh must be forced are now considered a bug. Please report."), TTR("OK"));
		} break;
		case SETTINGS_UPDATE_WHEN_CHANGED: {
			EditorSettings::get_singleton()->set("interface/editor/update_continuously", false);
			_update_update_spinner();
		} break;
		case SETTINGS_UPDATE_SPINNER_HIDE: {
			EditorSettings::get_singleton()->set("interface/editor/show_update_spinner", false);
			_update_update_spinner();
		} break;
		case SETTINGS_PREFERENCES: {
			editor_settings_dialog->popup_edit_settings();
		} break;
		case SETTINGS_EDITOR_DATA_FOLDER: {
			OS::get_singleton()->shell_show_in_file_manager(EditorPaths::get_singleton()->get_data_dir(), true);
		} break;
		case SETTINGS_EDITOR_CONFIG_FOLDER: {
			OS::get_singleton()->shell_show_in_file_manager(EditorPaths::get_singleton()->get_config_dir(), true);
		} break;
		case SETTINGS_MANAGE_EXPORT_TEMPLATES: {
			export_template_manager->popup_manager();
		} break;
		case SETTINGS_MANAGE_FBX_IMPORTER: {
#if !defined(ANDROID_ENABLED) && !defined(WEB_ENABLED)
			fbx_importer_manager->show_dialog();
#endif