Which hyponormal block Toeplitz operators are either normal or analytic?

We continue Curto–Hwang–Lee’s study of 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 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 extend Curto–Hwang–Lee’s result to block Toeplitz operators with symbols of bounded type. More precisely, we prove that if $\Psi ,\Psi ^{\ast }$ are matrix-valued functions of bounded type and the inner part of the Douglas–Shapiro–Shields factorization of $\Psi $ is a scalar inner function, then every hyponormal Toeplitz operator $T_{\Psi }$ whose square is also hyponormal must be either normal or analytic.