Saeedi Madani–Kiani regularity bound conjecture for binomial edge ideals

Let $G$ be a graph on $[n]$, let $S=K[x_1,\ldots,x_n,y_1,\ldots,y_n]$, and let $J_G$ be its binomial edge ideal. Let $c(G)$ denote the number of maximal cliques in $G$. Saeedi Madani–Kiani conjecture.

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

The source presents this as a conjecture proposed in [KM2] and proves the bound for closed graphs; the supplied text gives no resolution of the conjecture in general.