Nevo–Peeva conjecture on eventual linear resolutions of gapfree graph edge ideals

Let $G$ be a finite simple graph with vertex set $V(G)=\{1,\ldots,n\}$, let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$, and let $I(G)\subseteq S$ be the edge ideal generated by $x_ix_j$ for $\{i,j\}\in E(G)$. The graph $G$ is gapfree if any two disjoint edges have an edge meeting both of them. Nevo–Peeva's conjecture. If $G$ is gapfree, then $I(G)^q$ has linear resolution for all sufficiently large integers $q$. This conjecture is known in several special cases, including certain small-regularity and random-graph settings, but remains unresolved in general.