Which hyponormal block Toeplitz operators are either normal or analytic?

In this paper, we continue Curto-Hwang-Lee's work to study the connection between hyponormality and subnormality for block Toeplitz operators acting on the vector-valued Hardy space of the unit circle. Curto-Hwang-Lee's work focuses primarily on hyponormality and subnormality of block Toeplitz operators with rational symbols. By studying the greatest common divisor of matrix-valued inner functions and the ``weak" commutativity of matrix-valued inner functions, we extended Curto-Hwang-Lee's result to block Toeplitz operators with symbols of bounded type. More precisely, we proved that if $\Psi,\Psi^{\ast}$ are matrix-valued functions of bounded type and the inner part of $\Psi$ of Douglas-Shapiro-Shields factorization is a scalar inner function, then every hyponormal Toeplitz operator $T_{\Psi}$ whose square is also hyponormal must be either normal or analytic.