Rational approximation conjecture for ×(2,3)-circular Carathéodory functions

Let $\mathbb{D}$ be the unit disk, and let $\operatorname{Cara}(\mathbb{D};2,3)$ be the set of Carathéodory functions that are ×2- and ×3-circular, where ×$n$-circular means $\psi(z^n)=\frac1n\sum_{k=0}^{n-1}\psi(e^{2\pi i k/n}z)$ for every $z\in\mathbb{D}$. Rational approximation conjecture. Every element of $\operatorname{Cara}(\mathbb{D};2,3)$ is a compact-uniform limit of rational elements of $\operatorname{Cara}(\mathbb{D};2,3)$. The source presents this as a complex-analytic formulation related to the preceding rationality claim; it gives no evidence of resolution.