Local volume gap near the smooth point

Let $n$ be a positive integer, and let a smooth point mean a nonsingular $n$-dimensional klt singularity. Local volume gap conjecture. There exists $\varepsilon>0$ such that the only $n$-dimensional klt singularity $x\in X$ with

$$\widehat{\rm vol}(x,X)\geq(1-\varepsilon)n^n$$

is the smooth point. Since the smooth point has the largest local volume, this predicts a uniform gap below the maximum; the source presents it as a related open prediction.