Uniform delta-plt blowup conjecture for klt singularities

Let $n$ be a positive integer and let $\varepsilon>0$. A $\delta$-plt blowup of a klt singularity $x\in X$ is a plt blowup $Y\to X$ of a Kollár component $E$ such that $(Y,E)$ is $\delta$-plt, meaning $A_{Y,E}(F)>\delta$ for every prime divisor $F$ exceptional over $Y$. Uniform delta-plt blowup conjecture. There exists a constant $\delta=\delta(n,\varepsilon)>0$ such that every $n$-dimensional klt singularity $x\in X$ with

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

admits a $\delta$-plt blowup. The conjecture would provide uniformly controlled special degenerations and is stated as verified up to dimension three, but open in general.