Conjecture on Betti numbers of powers of closed binomial edge ideals

Let $G$ be a closed graph, let $J_G$ be its binomial edge ideal, and let $\ini_<(J_G)$ denote its initial ideal with respect to the specified term order. For every integer $i\geq 1$, the $i$-th powers satisfy

$$(\ini_<(J_G))^i=\ini_<(J_G^i),$$

and graded Betti numbers are taken for the corresponding graded ideals. Powers Betti-number conjecture. For every $i\geq 1$, the ideals $J_G^i$ and $(\ini_<(J_G))^i=\ini_<(J_G^i)$ have the same graded Betti numbers. This conjecture extends the earlier conjecture on the case $i=1$ and is motivated by computer experiments for small powers. The paper records supporting equalities for regularity in the closed-graph case and for depth when $J_G$ is Cohen–Macaulay, but the conjecture itself is not stated as resolved.