Subnormal $n$-roots of quasinormal operators are quasinormal

In a recent paper, Curto, R. E., Lee, S. H., Yoon, J. asked the following question:\textit{ Let $T$ be a subnormal operator, and assume that $T^2$ is quasinormal. Does it follow that $T$ is quasinormal?} In this paper, we give an affirmative answer to this question.