The Gibbs stability conjecture for Kähler–Einstein Fano manifolds

Let $X$ be a Fano manifold. A Kähler–Einstein metric on $X$ is denoted by $\omega_{KE}$, and let $\delta_N$ be the empirical measure of the corresponding canonical point process. The Gibbs stability conjecture. $X$ admits a unique Kähler–Einstein metric $\omega_{KE}$ if and only if $X$ is Gibbs stable. If $X$ is Gibbs stable, the empirical measures $\delta_N$ converge in probability to the normalized volume form of $\omega_{KE}$. This is proposed as a probabilistic analogue of the Yau–Tian–Donaldson conjecture; the parser supplies no evidence of resolution.