Monotonicity conjecture for the regularity of graph-power edge ideals

Let $G$ be a simple graph on vertex set $V(G)=[n]$, let $R=\Bbbk[x_1,\ldots,x_n]$, and let $I(G^d)$ be the edge ideal of the $d$-th power of $G$, where $d$ is a positive integer. The Castelnuovo–Mumford regularity is denoted by $\operatorname{reg}$. Monotonicity conjecture. For every simple graph $G$ and every positive integer $d$,

$$\operatorname{reg}\left(R/I(G^d)\right)\geq \operatorname{reg}\left(R/I(G^{d+1})\right).$$

The conjecture proposes that the regularity of edge ideals of successive graph powers is weakly decreasing. The paper proves this behavior for powers of forests and computes the regularity for powers of cycles, while the assertion for arbitrary simple graphs is suggested based on computations.