Strong asymptotics conjecture for Padé polynomials under Stahl's theorem

Let $f\in\mathscr H(\infty)$ and $f\in\mathscr A^0(\mathbb C\setminus\Sigma)$ for some finite set $\Sigma\subset\mathbb C$. Let $D=D(f)$ be Stahl's maximal domain associated with $f$, $S=S(f)=\partial D$ its Stahl compact set, and let $\mathfrak R_2=\mathfrak R_2(f)$ be the canonical hyperelliptic two-sheeted Riemann surface associated with $S$. Write $\mathbf z=z^{(1,2)}=(z,\pm)\in\mathfrak R_2$ for a point on this surface, and let $\Psi_n(\mathbf z)=\Psi_n(\mathbf z;f)$ be Nuttall's psi-function associated with $f$ and $\mathfrak R_2$. After a suitable normalization of the Padé polynomials $P_{n,j}(z)=P_{n,j}(z;f)$, $j=0,1$, and the remainder function $R_n$, the following relations hold in capacity inside $D$:

Strong asymptotics conjecture.

$$P_{n,j}(z)\overset{\operatorname{cap}}=\frac{(-1)^j}{f^j(z)}\Psi_n(z^{(1)})\bigl(1+o(1)\bigr),\qquad n\to\infty,$$ $$R_n(z)\overset{\operatorname{cap}}=\frac{\Psi_n(z^{(2)})}{w(z^{(2)})}\bigl(1+o(1)\bigr),\qquad n\to\infty,$$

where $w^2=H_{2g+2}(z)$, with $H_{2g+2}\in\mathbb C_{2g+2}[z]$, is the equation determining the hyperelliptic Riemann surface $\mathfrak R_2$ of genus $g$.

These relations describe the expected strong asymptotics of Padé polynomials and their remainder under the conditions of Stahl's theorem. In general, except in genus zero, such a representation need not be unique because of spurious zeros. The source presents this as a conjectural form rather than an established theorem.