Uniform boundedness conjecture for minimal log discrepancies of Kollár components

Fix $n\in\mathbb{N}$, $\varepsilon>0$, and let $I\subseteq [0,1]$ be a finite set. For a klt singularity $x\in (X,\Delta)$, let $\operatorname{mld}^{\mathrm{K}}(x,X,\Delta)$ denote its minimal log discrepancy computed among Kollár components, and let $\widehat{\operatorname{vol}}(x,X,\Delta)$ denote its local volume. Uniform boundedness conjecture. There exists a constant $A$, depending only on $n$, $\varepsilon$, and $I$, such that

$$\operatorname{mld}^{\mathrm{K}}(x,X,\Delta)\leq A$$

for every $n$-dimensional klt singularity $x\in (X,\Delta)$ satisfying $\operatorname{Coef}(\Delta)\subseteq I$ and $\widehat{\operatorname{vol}}(x,X,\Delta)\geq \varepsilon$.

This conjecture is motivated by the paper's boundedness criterion for K-semistable log Fano cone singularities and was already raised in the cited previous work; its general resolution is not supplied here.