The projective-dimension conjecture for edge rings with 2-linear resolutions

Let $G=(V,E)$ be a graph, let $n=|V|$, and let $k[G]=k[x_1,\ldots,x_n]/I(G)$ be its edge ring, where $I(G)$ is generated by $\{x_ix_j:\{i,j\}\in E\}$. Write $\operatorname{pd}(k[G])$ for the projective dimension of $k[G]$, and let the maximal degree of a vertex in $G$ mean the maximum of the vertex degrees. The edge-ring projective-dimension conjecture. If $k[G]$ has a 2-linear resolution, then

$$\operatorname{pd}(k[G])=\max_{v\in V}\deg_G(v).$$

The conjecture relates the homological complexity of an edge ring to the maximum vertex degree of its graph. In the supplied source, it is presented as the conjecture being treated; the provided material gives no resolution status beyond the paper’s title, so its database status is recorded as open.