Gross–Witten sum-rule conjecture for coupling below minus one

Let $\mu\in\mathcal M_1(\mathbb T)$ have Verblunsky coefficients $(\alpha_k)_{k\geq 0}\in\mathbb D^{\mathbb N}$. Let $\theta_{\tt g}$, $N^-$, $N^+$, $\lambda_i^+$, $\lambda_i^-$, and $\mathcal F_{\tt g}^{\pm}$ be the quantities associated with the Gross–Witten model, and let $\mathcal S_1^{\mathbb T}(\pi-\theta_{\tt g},\pi+\theta_{\tt g})$ denote the stated admissible class. For $\mu\in\mathcal S_1^{\mathbb T}(\pi-\theta_{\tt g},\pi+\theta_{\tt g})$ and any ${\tt g}<-1$, Gross–Witten sum-rule conjecture.

$$\mathcal K(\operatorname{GW}_{\tt g}\mid\mu)+\sum_{i=1}^{N^-}\mathcal F_{\tt g}^+(\lambda_i^+)+\sum_{i=1}^{N^+}\mathcal F_{\tt g}^-(\lambda_i^-)=H({\tt g})-{\tt g}\,\Re\left(\alpha_0-\sum_{k=1}^{\infty}\alpha_k\bar\alpha_{k-1}\right)-\sum_{k=0}^{\infty}\log(1-|\alpha_k|^2),$$

where $H$ is defined by the formula given in the source. If $\mu\notin\mathcal S_1^{\mathbb T}(\pi-\theta_{\tt g},\pi+\theta_{\tt g})$, the right-hand side is $+\infty$. This conjectures the extension of the Gross–Witten sum rule beyond the regime $|{\tt g}|\leq 1$; the supplied text does not state whether it has been proved or disproved.