Nazarov–Shapiro conjecture on weak asymptotic Toeplitzness of composition operators

Let $H^2$ be the Hardy space, let $\boldsymbol{D}$ be the unit disk, and let $C_{\varphi}$ denote the composition operator induced by a holomorphic self-map $\varphi$ of $\boldsymbol{D}$. An operator is weakly asymptotically Toeplitz (WAT) if the sequence $\{T_{\overline{z}}^m C_{\varphi}T_z^m\}_{m=0}^{\infty}$ converges in the weak operator topology; its limit is a Toeplitz operator, whose symbol is called the asymptotic symbol. Nazarov–Shapiro's conjecture. If $\varphi$ is neither a rotation nor the identity map, then $C_{\varphi}$ is WAT with asymptotic symbol zero. This conjecture concerns the weak asymptotic Toeplitzness of composition operators and extends results known for several classes of symbols; the supplied text does not state whether it has been resolved.