Conjectured asymptotics for Hermite–Padé polynomials with a nonreal symmetric condenser plate

Let $f\in\mathbb C(z,w)$ and $f_\infty\in\mathscr H(\infty)$. Assume the setting of the subsection in which $E\subset\mathbb R$, $E\cap F=\varnothing$, and $F$ is symmetric with respect to the real line, meaning $z\in F$ if and only if $\overline z\in F$; the endpoints satisfy the stated conditions, including $A_j<B_j$ in the real case and $A_j=\overline{B_j}\notin\mathbb R$ in the nonreal case. General-condenser asymptotics conjecture. Under these assumptions, the relations (8)–(9) and (3.2)–(3.4) are valid as $n\to\infty$.

This proposes extending the paper’s asymptotic theory from condensers whose two plates lie on the real line to the general Nuttall condenser with a real plate $E$ and a conjugation-symmetric plate $F$. The source does not indicate that the assertion has been proved or refuted.