Hibi–Matsuda–Tsuchiya conjecture on linear resolutions of edge rings

Let $G$ be a finite connected simple graph, let $K[G]$ denote its edge ring over a field $K$, and let $q\geq 3$. A graded ring has a $q$-linear resolution when its minimal graded free resolution is $q$-linear. Hibi–Matsuda–Tsuchiya conjecture. If $K[G]$ has a $q$-linear resolution, then $K[G]$ is a hypersurface. The case $q=3$ was proved by Hibi, Matsuda, and Tsuchiya; the assertion for general $q\geq 3$ remains unresolved.