The lower semicontinuity conjecture for minimal log discrepancies

Let $(X,B)$ be a log variety, and let $a(x)=a(x;X,B)$ denote the minimal log discrepancy at a closed point $x\in X$. Lower semicontinuity conjecture. For every closed point $x\in X$, there exists a neighborhood $x\in U\subseteq X$ such that

$$a(x;X,B)=\inf_{x'\in U}a(x';X,B).$$

Equivalently, the function of minimal log discrepancies is lower semicontinuous. The source explains that this conjecture implies the first part of the maximal minimal-log-discrepancy conjecture and notes that the function is constant equal to $\dim X$ on an open dense subset; the supplied text gives no resolution status.