Strong Mabuchi energy expansion conjecture for beta-ensembles on the Riemann sphere

Let $\phi$ be a function on $\mathbb{C}$ with logarithmic growth and finite relative entropy $D_{\mu_0}(\mu_{\phi})$, where $\mu_0:=e^{-2\psi_0}/\pi$ is the standard probability measure on the Riemann sphere $\mathbb{P}^1$. Set $p:=2/\beta-1$, and let $D^{(N)}$, $\lambda$, $\mathcal{F}$, $\mathcal{M}$, and $\xi_\beta$ have the meanings used in the source.

Strong beta-ensemble expansion conjecture.

$$\frac{1}{N(N+p)\beta}\log\int_{\mathbb{C}^{N}}\left(|D^{(N)}|^{2}e^{-(N+p)\phi}\right)^{\beta}d\lambda^{\otimes N}=-\mathcal{F}(\phi)+\frac{\log N}{2}N^{-1}-N^{-1}\left(\left(\frac{1}{\beta}-\frac{1}{2}\right)\mathcal{M}(P\phi)-\xi_{\beta}\right)+o(1).$$

This is presented as the strongest-form conjecture for beta-ensembles on the Riemann sphere, with Mabuchi's K-energy functional for $\mathcal{O}(1)\rightarrow\mathbb{P}^1$ appearing in the next-order term.