Lauritzen's score matching estimability conjecture

Let $G=(V,E)$ be a graph with $m$ vertices, and let $n$ be the sample dimension. The graph is $n$-estimable when its score matching estimator exists with probability one.

Lauritzen's conjecture. The graph $G$ is $n$-estimable if and only if

$$\#V+\#E\leq nm-{n\choose 2}.$$

This conjecture proposed a purely numerical characterization of the score matching threshold. It is refuted by a graph counterexample in the paper.