Existence of plt blow-ups from positive local volume

Let $n\geq 2$ be a positive integer and let $\eta,\epsilon$ be positive real numbers. Let $x\in (X,\Delta)$ be an $n$-dimensional klt singularity, with

$$\Delta=\sum_{i=1}^m a_i\Delta_i,$$

where each $\Delta_i\geq 0$ is a Weil divisor. Let $\widehat{\operatorname{vol}}(x,X,\Delta)$ denote its local volume. Conjecture on plt blow-ups. There exists a positive real number $\delta$, depending only on $n,\eta$ and $\epsilon$, such that if $a_i>\eta$ for every $i$ and $\widehat{\operatorname{vol}}(x,X,\Delta)>\epsilon$, then $x\in(X,\Delta)$ admits a $\delta$-plt blow-up.

This is the converse to the paper’s proved lower-bound direction relating positive local volume to singularities admitting plt blow-ups. The conjecture is confirmed in the special cases treated by the paper, while it is open in general.