Depth-seven conjecture for edge rings of finite graphs

Let $G$ be a finite graph on the vertex set $[d]=\{1,\ldots,d\}$ with $d\geq 7$, and let $K[G]$ be its edge ring. Depth-seven conjecture.

$$\operatorname{depth} K[G]\geq 7.$$

The claim arises from computations showing that connected nonbipartite graphs with seven vertices have edge rings of depth at least seven. The preceding discussion presents it as a temptation rather than an established theorem; the supplied status gives no evidence of resolution.