Normality and Cohen–Macaulayness conjecture for Gaussian Bayesian network ideals

Let $G$ be a graph for a Gaussian Bayesian network, let $I_G$ be its vanishing ideal, and let $4\mathbb{C}[\Sigma]4$ be the polynomial ring in the covariance coordinates.

Normality and Cohen–Macaulayness conjecture. The quotient ring

$$\mathbb{C}[\Sigma]/I_G$$

is normal and Cohen–Macaulay for all $G$.

The conjecture was verified computationally for all graphs on at most five vertices and for six-vertex graphs with fewer than eight edges. It is proved in the paper when the underlying graph is a tree, while the general assertion remains open.