Alilooee–Banerjee–Beyarslan–Hà regularity conjecture for powers of edge ideals

Let $G$ be a graph, let $I(G)$ be its edge ideal, and let ${\rm reg}(I(G))$ denote its Castelnuovo–Mumford regularity. For each integer $s\geq 1$, Alilooee–Banerjee–Beyarslan–Hà's conjecture.

$${\rm reg}(I(G)^s)\leq 2s+{\rm reg}(I(G))-2.$$

This is a stronger form of the conjecture that the constant term in the eventual linear function ${\rm reg}(I(G)^s)$ is at most ${\rm reg}(I(G))-2$. It is known for several classes of graphs, including graphs with chordal complement, but remains open for arbitrary graphs.