The componentwise linear symbolic power conjecture for linear-resolution edge ideals

Let $G$ be a finite simple graph, let $S=K[x_1,\ldots,x_n]$ be the standard graded polynomial ring over a field $K$, and let $I(G)$ be its edge ideal. For a positive integer $k$, let $I(G)^{(k)}$ denote the $k$th symbolic power. An ideal is componentwise linear if each of its graded components has a linear resolution.

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

By Fröberg's characterization, the hypothesis is equivalent to $G$ being cochordal. The conjecture is proved for several families of cochordal graphs, including those whose complements are block graphs, proper interval graphs, or specified chordal graphs, but is open in general.