The ODP conjecture on the second largest local volume of klt singularities

Let $x\in X$ be an $n$-dimensional klt singularity, and let $\widehat{\operatorname{vol}}(x,X)$ denote its local volume, defined as the infimum of the normalized volumes of valuations centered at $x$. An ordinary double point is a klt singularity analytically equivalent to a quadratic hypersurface singularity of ordinary double-point type. ODP conjecture. The second largest local volume of an $n$-dimensional klt singularity is

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

Moreover,

$$\widehat{\operatorname{vol}}(x,X)=2(n-1)^n$$

if and only if $x\in X$ is an ordinary double point. The maximal local volume is known to be $n^n$, attained exactly at smooth points; this conjecture predicts the next possible value and characterizes the singularities attaining it.