5 problems

Let $G$ be a chordal graph, and let $J(G)$ be its cover ideal. A graded ideal is componentwise linear when each ideal generated by its elements of a fixed degree has a linear resol…

Soleyman Jahan–Zheng conjecture. If $I$ has componentwise linear quotients, then $I$ has linear quotients.

Componentwise linear symbolic power conjecture. If $I(G)$ has a linear resolution, then $I(G)^{(k)}$ is componentwise linear for every $k\geq 1$.

Cochordal graph conjecture. All symbolic powers $I(G)^{(k)}$ are componentwise linear whenever $G$ is cochordal, that is, whenever the complementary graph $G^c$ is chordal.

Ha and Tuyl's problem. Is $J(\Delta)^s$ componentwise linear for all $s\geq 1$?