Minimal-cycle Betti-number conjecture for graph curves

Let $G$ be a graph on $d$ vertices, embedded as in Theorem 1.3. Let $n$ denote the girth of $G$, and suppose that $d=2g+1+p$ and $n-2\leq p$. Minimal-cycle Betti-number conjecture. The property $N_{2,p}$ fails, and $\beta_{n-2,n}$ is equal to the number of cycles of length $n$ in $G$. This predicts that a graded Betti number records the number of shortest cycles in the graph under the stated numerical hypotheses.