Curto–Hwang–Lee conjecture on hyponormal block Toeplitz operators

Let $\Phi \in L^{\infty}(\mathbb{T}, M_n)$, and let $T_\Phi$ be the block Toeplitz operator with symbol $\Phi$. Suppose that $T_\Phi$ is hyponormal and that its self-commutator $[T_\Phi^{\ast},T_\Phi]$ has finite rank. Let $\mathcal{E}(\Phi)$ denote the Cowen set of $\Phi$. A finite Blaschke–Potapov product is a matrix-valued inner function of finite degree.

Curto–Hwang–Lee conjecture. There exists a finite Blaschke–Potapov product $B\in\mathcal{E}(\Phi)$ such that

$$\operatorname{rank}[T_\Phi^{\ast},T_\Phi]=\deg(\det B).$$

This conjecture predicts that the rank of the self-commutator is determined by the degree of the determinant of a finite Blaschke–Potapov product in the Cowen set. Its resolution is not indicated in the supplied text.