Domain and logarithmic-potential description of asymptotic zero measures

Let $varPsi(z)=\operatorname{Max}\{\widetilde H_i(z)\}$, let $K$ be the support of $triangle\varPsi$, and let $mu$ be an asymptotic measure associated with the hypergeometric polynomials in the stated class. Write $L_mu$ for the logarithmic potential of $mu$. Global potential conjecture. There exists a connected and simply connected domain $D\subset\mathbb C$ such that, with $K'=K\cap D$, $K'$ is the support of $mu$ and $L_mu=varPsi$ in $D$; moreover, $L_mu=H_1$ outside $D$. This gives a global description of which part of the level-curve set supports the asymptotic measure, complementing the local statement that the support lies along some level curves.