Bolognini–Macchia–Strazzanti's Cohen–Macaulayness conjecture for binomial edge ideals

Let $G$ be a graph, and let $J_G$ denote its binomial edge ideal. The graph $G$ is accessible if its cutsets form an accessible system.

Bolognini–Macchia–Strazzanti's conjecture. $J_G$ is Cohen–Macaulay if and only if $G$ is accessible.

This conjecture proposes a combinatorial characterization of Cohen–Macaulay binomial edge ideals. The source presents it as a conjecture of Bolognini, Macchia, and Strazzanti; its resolution is not specified here.