Rationality conjecture for extreme ×(2,3)-circular Carathéodory functions

Let $\mathbb{D}$ be the unit disk, and let $\operatorname{Cara}(\mathbb{D};2,3)$ be the set of holomorphic functions $\psi:\mathbb{D}\to\mathbb{C}$ with $\operatorname{Re}(\psi(z))>0$, $\psi(0)=1$, and $\psi(z^n)=\frac1n\sum_{k=0}^{n-1}\psi(e^{2\pi i k/n}z)$ for $n=2,3$. Call such a function extreme if it is an extreme point of this convex set. Rationality conjecture. Every extreme element of $\operatorname{Cara}(\mathbb{D};2,3)$ is rational. The source presents this as one of three complex-analytic formulations equivalent to the ergodic-theoretic conjectures; it gives no evidence of resolution.