The Fekete–Pommerenke energy conjecture for Coulomb gases on Jordan curves

Let $\eta$ be a Jordan curve of unit capacity, and let $Z_{n,\infty}(\eta)$ denote the zero-temperature (Fekete-point) partition function.

Fekete–Pommerenke energy conjecture.

$$\varlimsup_{n\to\infty}8\log\frac{Z_{n,\infty}(\eta)}{Z_{n,\infty}(\mathbb{T})}<\infty$$

if and only if $\eta$ is a Weil–Petersson quasicircle; in that case, the limit exists and equals the Fekete–Pommerenke energy $I^F(\eta)$.

This conjecture is the zero-temperature analogue of the finite-$\beta$ characterization and connects Fekete-point asymptotics with the finiteness of $I^F$. The source gives no resolution.