Gamma-function singularity conjecture for beta-ensemble free energies

Let $V_{d+1}(x)$ be any potential of degree $d+1$, and let ${\widetilde{\epsilon}}_i$ be the occupation numbers on the cycles ${\widetilde{{\cal A}}}_i$, for $i=1,\ldots,d$. Here $[\int\log\Gamma](x)$ denotes an antiderivative of $\log\Gamma(x)$. Gamma-function singularity conjecture. The singular part of ${\mathcal F}_0$ has the form

$$\operatorname{sing}{\mathcal F}_0=\hbar^2\sum_{i=1}^d\frac{1}{2}[\int\log\Gamma]({\widetilde{\epsilon}}_i/\hbar).$$

This conjecture proposes a universal description of the singular part of the genus-zero free energy in terms of the occupation numbers of the cycles. The preceding analysis derives the corresponding one-cycle behavior and matches the expected large-positive-argument Gaussian matrix-model asymptotics, but the general statement is not resolved here.