Madani–Kiani's generalized regularity bound for connected graphs

Let $G$ be a connected graph on vertex set $[n]$, let $K_m$ be the complete graph on $[m]$, let $J_{K_m,G}$ be the generalized binomial edge ideal, let $c(G)$ be the number of maximal cliques of $G$, and assume $m<n$. Madani–Kiani's connected-graph conjecture.

$$\operatorname{reg}(S/J_{K_m,G})\leq \min\{(m-1)c(G),n-1\}.$$

The source says that this bound was proved for chordal graphs, but does not provide a general proof or disproof for connected graphs.