분사구문에서 though를 남겨도 되나요?

“Though he was rich, the millionaire was not rich”를 분사구문으로 만들면 “Being rich, the millionaire was not rich”가 되나요?
접속사의 뜻을 분명히 하려고 남길 수 있다고 배웠는데, “Though being rich, the millionaire was not rich”도 맞는 문장인지 궁금합니다.