Lower semi-continuity conjecture for generalized minimal log discrepancies

Let $(X,\Delta+M)/Z$ be a generalized pair. Define

$$m:|X|_{\mathrm{cl}}\to\mathbb{R}\cup\{-\infty\},\qquad x\mapsto\operatorname{mld}_x(X,\Delta+M),$$

where $|X|_{\mathrm{cl}}$ denotes the set of all closed points of $X$ with the Zariski topology. Lower semi-continuity conjecture for generalized minimal log discrepancies. The function $m$ is lower semi-continuous. The paper reduces this generalized statement to the corresponding statement for usual log pairs and proves it in several cases, including dimension at most three, smooth varieties, locally complete intersection singularities, and quotient singularities; the general case remains open.