The projective-dimension bound for binomial edge ideals of high regularity

Let $G$ be a graph on $n\ge 7$ non-isolated vertices. Suppose that $\operatorname{reg}(J_G)=n-1$. Projective-dimension bound. Then

$$\operatorname{proj\,dim}(J_G)\le n.$$

The paper reports that this stronger bound is suggested by computational experiments, while the stated general result only gives $\operatorname{proj\,dim}(J_G)\le 2n-7$ for $n\ge 6$; the conjectured bound remains open.