Periodic equidistribution conjecture for rational extreme Carathéodory functions

Let $\mathbb{D}$ be the unit disk, and let $\operatorname{Cara}(\mathbb{D};2,3)$ be the set of ×2- and ×3-circular Carathéodory functions; call an element rational extreme when it is both rational and extreme in this convex set, and let the number of its poles be counted with multiplicity. Periodic equidistribution conjecture. If $(\psi_n)_{n\in\mathbb{N}}$ is a sequence of rational extreme elements of $\operatorname{Cara}(\mathbb{D};2,3)$ whose number of poles tends to infinity, then $(\psi_n)$ converges compact-uniformly to the constant function $1$. The source gives no evidence of resolution.