Herzog–Hibi–Ohsugi conjecture on powers of cover ideals of chordal graphs

Let $G$ be a finite simple graph, and let $J(G)$ denote its cover ideal. A graph is chordal if it has no induced cycle of length at least $4$. An ideal is componentwise linear if, for every non-negative integer $d$, the ideal generated by its homogeneous elements of degree $d$ has a linear resolution.

Herzog–Hibi–Ohsugi conjecture. If $G$ is chordal, then

$$J(G)^s$$

is componentwise linear for all $s \geq 1$.

The conjecture asks whether the componentwise linearity known for cover ideals of chordal graphs is preserved under all ordinary powers. The source presents this as an open conjecture proposed by Herzog, Hibi, and Ohsugi.