The WAT conjecture on the torus

Let $\phi$ be a bounded holomorphic function on the unit disc $\mathbb{U} \subset \mathbb{C}$, let $k$ be an integer, and define the sequence $\{ c_n \}$ by $c_n= \int_{\partial \mathbb{U}} \phi(z)^n z^{k-n} dz$. Nazarov and Shapiro conjectured that if $\Vert \phi \Vert _{L^\infty(\mathbb{U})} \leq 1$ and $\phi$ is not a rotation, then $\lim_{n \to \infty} c_n = 0$. A consequence of this conjecture would be that any composition operator is weakly asymptotically Toeplitz on the Hardy space $H^2(\mathbb{U})$. We formulate a higher-dimensional version of this conjecture and use Fourier analytic techniques to obtain results that improve on what is currently known in dimension one.