Pitones–Reyes–Toledo conjecture on Cohen–Macaulay vertex-weighted edge ideals

Let $\mathcal{D}=(V(\mathcal{D}),E(\mathcal{D}))$ be a vertex-weighted oriented graph, let $G$ be its underlying graph, and let $S=\mathbb{K}[x_1,\ldots,x_n]$ be the polynomial ring on the vertices over a field $\mathbb{K}$. Write $I(\mathcal{D})$ for the vertex-weighted edge ideal of $\mathcal{D}$ and $I(G)$ for the usual edge ideal of $G$. Pitones–Reyes–Toledo conjecture. If $I(\mathcal{D})$ is unmixed and $S/I(G)$ is Cohen–Macaulay, then $S/I(\mathcal{D})$ is Cohen–Macaulay.

This conjecture asks whether Cohen–Macaulayness of the underlying graph, together with unmixedness of the vertex-weighted edge ideal, guarantees Cohen–Macaulayness of the weighted ideal. The paper presents it as a conjecture of Pitones, Reyes and Toledo; no resolution is supplied in the given text.