Conjecture on commutator rank and bidegrees of matrix-valued inner functions

Let $\Theta$ be a $d\times d$ matrix-valued inner function on the bidisk, and let $S_{z_1}$ denote the associated compressed shift operator. Write $\deg_1\Theta$ and $\deg_2\det\Theta$ for the respective degrees in the two variables, and let $n$ be the commutator rank. Commutator-rank conjecture.

$$\operatorname{Rank}[S_{z_1},S_{z_1}^*]=n$$

if and only if $\deg_1\Theta\leq 1$ and $\deg_2\det\Theta=n$. This conjecture would extend the corresponding scalar result to the matrix-valued setting; the paper establishes only partial implications, and the general equivalence remains open because scalar arguments do not directly generalize.