Regularity bound by the number of maximal cliques for binomial edge ideals

Let $G$ be a graph, let $S={\mathbb K}[x_1,\ldots,x_n,y_1,\ldots,y_n]$ be the polynomial ring over a field ${\mathbb K}$, and let $J_G\subseteq S$ be its binomial edge ideal. Let $c(G)$ denote the number of maximal cliques of $G$, and let

$$\operatorname{reg} S/J_G=\max\{j-i:\beta_{i,j}(S/J_G)\neq 0\}.$$

The clique-count regularity conjecture. For every graph $G$,

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

The bound is known for closed, or proper interval, graphs. The source presents it as a conjecture posed in 2013, and the supplied material gives no resolution for arbitrary graphs.