Hibi–Matsuda's h-polynomial bound for binomial edge ideals

Let $G$ be a graph on $n$ vertices, let $J_G$ be its binomial edge ideal in a polynomial ring $S$, and let $h_{S/J_G}(t)$ be the numerator polynomial of the Hilbert series of $S/J_G$. Hibi–Matsuda's conjecture.

$$\operatorname{reg}(S/J_G)\leq \deg h_{S/J_G}(t).$$

The source attributes this conjecture to Hibi and Matsuda and does not provide evidence that it has been resolved in general.