Madani–Kiani's maximal-clique regularity conjecture for binomial edge ideals

Let $G$ be a simple graph, let $S$ be the polynomial ring used to define its binomial edge ideal $J_G$, and let $c(G)$ be the number of maximal cliques of $G$. Madani–Kiani's conjecture.

$$\operatorname{reg}(S/J_G)\leq c(G).$$

The conjecture was proved for closed graphs, generalized block graphs, chordal graphs, fan graphs of complete graphs, $P_4$-free graphs, block graphs, and semi-block graphs, among other classes. The source states that it was subsequently proved for all graphs with a sharp bound, so the conjecture is resolved.