Capacity convergence conjecture for the second Hermite approximant

Let $f$ belong to the Laguerre class $\mathscr L$, represented by

$$f(z)=\prod_{j=1}^q\left(\frac{z-e_{2j-1}}{z-e_{2j}}\right)^{\alpha_j},$$

where $\alpha_j\in\mathbb R\setminus\mathbb Z$, $e_j\in\mathbb R$, and $e_1<\dots<e_{2q}$. Assume that $1,f,f^2$ are rationally independent over $\mathbb C(z)$. Let $Q_{n,0}$ and $Q_{n,2}$ be the corresponding type I Hermite–Padé polynomials, and let $F=\bigcup_{j=1}^mF_j$ be a compact set consisting of finitely many closed analytic arcs. Capacity convergence conjecture. For $z\in\mathbb C\setminus F$,

$$\frac{Q_{n,0}}{Q_{n,2}}(z)\overset{\operatorname{cap}}\longrightarrow f^2(z),\qquad n\to\infty.$$

This conjecture predicts convergence in capacity of the relevant Hermite approximants outside a finite union of analytic arcs; its status is not resolved in the supplied source.