Max-min compact-set conjecture for Tschebyshev–Padé approximants

Let $F\in\mathcal F_f$ be an admissible compact set for $f$, and let $\{\Omega_n\}$ be an interpolation table disjoint from $F$. Assume that the limiting density $\omega$ in the interpolation-table limit exists and that its support is disjoint from $e$. For the denominators $Q_n$, write $\nu(Q_n)$ for their zero-counting measure, and let $\lambda_{\widetilde\Gamma}$ denote the equilibrium measure associated with a compact set $\widetilde\Gamma$. Max-min compact-set conjecture. There exists a unique compact set $\widetilde\Gamma$ having the max-min property, such that

$$\frac{1}{n}\nu(Q_n)\overset{*}{\to}\lambda_{\widetilde\Gamma}\qquad\text{as }n\to\infty.$$

The theorem preceding this conjecture proves the claim under more restrictive assumptions through the max-min energy and GRS methods; the authors state that extending it to general limiting densities requires modifications whose validity is not yet formally established, and they note that further restrictions on $\omega$ may be necessary.