Matsuda–Murai's characterization of maximal regularity for binomial edge ideals

Let $G$ be a finite simple graph, with $n(G)$ denoting its number of vertices and $J_G$ its binomial edge ideal in $S=K[x_i,y_i:i\in V(G)]$. Matsuda–Murai's conjecture.

$$\operatorname{reg}(S/J_G)=n(G)-1$$

if and only if $G$ is a path graph on $n(G)$ vertices. The claim concerns equality in the general upper bound for binomial edge ideals; the source does not state whether it has been resolved.