Regularity bound by the h-polynomial degree for binomial edge ideals

Let $G$ be a finite simple graph on the vertex set $[n]=\{1,\ldots,n\}$, let $K[\boldsymbol{x},\boldsymbol{y}]=K[x_1,\ldots,x_n,y_1,\ldots,y_n]$ be the standard graded polynomial ring over a field $K$, and let $J_G$ be the binomial edge ideal generated by $x_i y_j-x_j y_i$ for the edges $\{i,j\}$ of $G$. Write

$$h_{K[\boldsymbol{x},\boldsymbol{y}]/J_G}(\lambda)=h_0+h_1\lambda+\cdots+h_s\lambda^s$$

for the $h$-polynomial of $K[\boldsymbol{x},\boldsymbol{y}]/J_G$, and let $\operatorname{reg}$ denote Castelnuovo--Mumford regularity. Regularity–$h$-degree conjecture. One has

$$\operatorname{reg}(K[\boldsymbol{x},\boldsymbol{y}]/J_G)\leq \deg h_{K[\boldsymbol{x},\boldsymbol{y}]/J_G}(\lambda).$$

Equality is known when the quotient is Cohen--Macaulay, but the inequality is supported here only by computational experience and remains open in general.