The hypersurface conjecture for edge rings with q-linear resolutions

Let $G$ be a finite connected simple graph, and let $K[G]$ denote its edge ring. A graded ring has a $q$-linear resolution when its minimal graded free resolution is $q$-linear, and assume $q\geq 3$. Edge-ring hypersurface conjecture. If $K[G]$ has a $q$-linear resolution, then $K[G]$ is a hypersurface. The preceding theorem establishes the claim for $q=3$; the conjecture asks whether it holds for every $q\geq 3$.