Multiplicity and embedding-dimension boundedness conjecture

Let $n$ be a positive integer, let $\varepsilon>0$, and let $\operatorname{mult}_xX$ denote the multiplicity and the embedded dimension of $x\in X$ its embedding dimension. Multiplicity and embedding-dimension boundedness conjecture. There exists a constant $M=M(n,\varepsilon)$ such that every $n$-dimensional klt singularity $x\in X$ with

$$\widehat{\rm vol}(x,X)\geq\varepsilon$$

has

$$\operatorname{mult}_xX\leq M$$

and embedded dimension at most $M$. This is described as a consequence expected from the Boundedness Conjecture, because these invariants are bounded in bounded families and non-decreasing under specialization; no resolution is stated.