Conjecture on numerical ranges of composition operators with elliptic symbols

Let $D$ be the unit disk, let $H^2(D)$ be the Hardy space on $D$, and let $C_\varphi$ denote the composition operator induced by an elliptic automorphism $\varphi$ of finite order $p$. Assume that the fixed point of $\varphi$ is not $0$.

Numerical-range conjecture. The numerical range of $C_\varphi$ on $H^2(D)$ is the interior of the convex hull of an algebraic curve of class $p$ and degree $p^2-p$. Moreover, the real foci of the curve are exactly the eigenvalues of $C_\varphi$ on $H^2(D)$, namely

$$\{e^{2k\pi i/p}\}_{k=1}^p.$$

This conjecture proposes a description of the numerical range in the higher-order elliptic case, extending the observed quadratic-curve behavior when the order is smaller. The source gives no resolution, so its status is open.