Boundedness of minimal log discrepancies of Kollár components with positive local volume

Let $n\in\mathbb{N}^*$, let $\varepsilon>0$, and let $I\subseteq [0,1]\cap\mathbb{Q}$ be a finite set. For an $n$-dimensional klt pair $(X,\Delta)$ with $\operatorname{Coef}(\Delta)\subseteq I$ and a point $\eta\in X$, not necessarily closed, assume

$$\widehat{\operatorname{vol}}(\eta,X,\Delta)\geq \varepsilon.$$

Boundedness conjecture. There exists a constant $A=A(n,\varepsilon,I)$ such that

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

Here $\operatorname{mld}^{\mathrm{K}}$ is the smallest log discrepancy among Kollár components over $\eta$. The paper proves this in dimensions at most three and shows that, in any dimension, it implies the boundedness conjecture for klt singularities with normalized volume bounded below.