Ambro's lower semi-continuity conjecture for minimal log discrepancies

Let $(X,\mathfrak{a})$ be a log pair, and let $|X|$ denote the set of all closed points of $X$ with the Zariski topology. The function

$$|X| \to \mathbb{R}_{\ge 0} \cup \{-\infty\};\quad x \mapsto \operatorname{mld}_x(X,\mathfrak{a})$$

is the function of minimal log discrepancies on the closed points of $X$.

Lower semi-continuity conjecture. The function $x\mapsto \operatorname{mld}_x(X,\mathfrak{a})$ is lower semi-continuous.

Ambro proposed this conjecture for minimal log discrepancies. It is known in several cases, including when $\dim X=3$, when $X$ is smooth, for normal local complete intersection varieties, and for quotient singularities; the general conjecture remains open.