The K-stability–Gibbs stability equivalence conjecture

Let $X$ be a Fano manifold. The K-stability–Gibbs stability conjecture. $X$ is (uniformly) K-stable if and only if $X$ is (uniformly) Gibbs stable. Here uniform Gibbs stability means that there exists $\epsilon>0$ such that, for sufficiently large $k$, the log canonical threshold of the divisor $\mathcal{D}_{N_k}$ satisfies $\operatorname{lct}(\mathcal{D}_{N_k})>1+\epsilon$. This is a proposed algebro-geometric analogue relating the two stability notions; no resolution evidence is supplied.