Schatten-class characterization for truncated Hankel operators of one-component inner functions

Let $\theta$ be a one-component inner function, let $\mathcal K_\theta=H^2\ominus\theta H^2$, and let $\Gamma_{\phi}:\mathcal K_\theta\to\overline{z\mathcal K_\theta}$ be a truncated Hankel operator with standard symbol $\phi\in\overline{\mathcal K_{\theta^2}\cap zH^2}$. Let $A_{\theta\phi}:\mathcal K_\theta\to\mathcal K_\theta$ be the associated truncated Toeplitz operator, let $C_\phi:L^2(\mu)\to L^2(\mu_\theta)$ be the associated truncated Hankel operator between the Clark-measure spaces, let $B_p$ be the classical Besov space on the unit circle, and let $B_p(\mu_\theta)$ be the Besov space defined using mean oscillations over the dyadic subarcs determined by the support of the Clark measure $\mu_\theta$. The Schatten-class conjecture. For every $p\in(0,\infty)$, the following assertions are equivalent: (1) $\Gamma_\phi\in S^p$; (2) $A_{\theta\phi}\in S^p$; (3) $C_\phi\in S^p$; (4) $\Gamma_\phi=\sum_n a_n\Gamma_{\phi_n}$ for rank-one operators $\Gamma_{\phi_n}$ of unit norm and coefficients satisfying $\sum_n|a_n|^p<\infty$; (5) $\phi\in B_p+\overline{\theta^2H^2}+H^2$; and (6) $\phi\in B_p(\mu_\theta)$. This conjecture seeks a complete and equivalent description of Schatten membership for truncated Hankel operators, relating operator ideals, rank-one decompositions, and Besov regularity; its status is unresolved in the supplied source.