K-reduced partition function and empirical measure conjecture

Let $(X,\Delta)$ be a log Fano manifold admitting a Kähler–Einstein metric. Let $Z_{\boldsymbol{m}_{N}}$ be the K-reduced partition function, let $\mathcal{M}$ be the Mabuchi functional, and let $\delta_N$ be the empirical measure under the K-reduced probability measure $\mu_{\boldsymbol{m}_N}^{(N)}$. K-reduced partition function conjecture.

$$-\lim_{N\to\infty}N^{-1}\log Z_{\boldsymbol{m}_{N}}=\inf\mathcal{M}.$$

Consequently, $Z_{\boldsymbol{m}_{N}}<\infty$ for sufficiently large $N$ and

$$\lim_{N\to\infty}\delta_N=\mu_{\mathrm{KE}}$$

in probability, where $\mu_{\mathrm{KE}}\in\mathcal{P}(X)_0$ is the normalized volume form of the unique Kähler–Einstein metric with vanishing moment. This predicts both the asymptotic free energy and convergence of the conditioned point process; the source provides no resolution.