The linear quotients conjecture for symbolic powers of 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.

Linear quotients conjecture for symbolic powers. If $I(G)$ has a linear resolution, then $I(G)^{(k)}$ has linear quotients for every $k\geq 1$.

This is presented as a more general statement than the componentwise linearity conjecture, since linear quotients imply componentwise linearity. It is motivated by the paper's results for several cochordal graph families and remains open in general.