Saeedi and Kiani's regularity bound by maximal cliques

Let $G$ be a graph, let $S={\mathbb K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$, and let $J_G$ be its binomial edge ideal. Denote by $c(G)$ the number of maximal cliques of $G$. Saeedi and Kiani's conjecture. The Castelnuovo–Mumford regularity satisfies

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

The bound is known for closed graphs, and the paper proves it for chordal graphs. The statement is presented as a conjecture for arbitrary graphs in the supplied text, with no resolution beyond the chordal case established there.