Matrix Gross–Witten sum-rule conjecture

Let $\Sigma\in\mathcal M_{p,1}(\mathbb T)$ have infinite support, and let $(\boldsymbol\alpha_k)_{k\geq0}\in(\mathbb B_p)^{\mathbb N_0}$ be its matrix Verblunsky coefficients. Let $T(\boldsymbol\alpha_0,\dots)$ be the matrix expression specified in the source, and let $H_p({\tt g})$ be a constant. Matrix Gross–Witten sum-rule conjecture. If $|{\tt g}|\leq1$, then

$$\mathcal K(\mathbf 1\cdot\operatorname{GW}_{-{\tt g}}\mid\Sigma)=\sum_{k=0}^{\infty}-\log\det(\mathbf 1-\boldsymbol\alpha_k\boldsymbol\alpha_k^\dagger)+{\tt g}\,\Re\,\operatorname{tr}T(\boldsymbol\alpha_0,\dots)+H_p({\tt g}).$$

If $|{\tt g}|>1$, a similar identity holds with the additional term $\sum_{i=1}^{N^-}\mathcal F^+_{-{\tt g}}(\lambda_i^-)+\sum_{n=1}^{N^+}\mathcal F^-_{-{\tt g}}(\lambda_i^+)$ on the left-hand side. This is presented as the matrix analogue of the scalar Gross–Witten sum rules, but the supplied text gives no resolution status.