The Gorensteinness conjecture for rings in $\mathbf{G}(r)$

Let $R$ be a local ring in the class $\mathbf{G}(r)$, where $r\ge2$. Gorensteinness conjecture. If $R$ is in $\mathbf{G}(r)$ for some $r\ge2$, then $R$ is Gorenstein. The only known examples in $\mathbf{G}(r)$ are the Gorenstein rings; the conjecture proposes that these are the only possibilities.