Ordinary double point volume gap conjecture

Let $n$ be a positive integer, and let an ordinary double point denote the corresponding $n$-dimensional klt quadratic singularity. Ordinary double point volume gap conjecture. The second largest local volume of an $n$-dimensional klt singularity is

$$2(n-1)^n,$$

and this value is achieved only by the ordinary double point. This refines the expected discreteness of local volumes and would have consequences for K-moduli of cubic hypersurfaces; the source says that even the existence of a volume gap appears nontrivial.