Burnham–Rosen–Sidman–Vermeire conjecture on graph curves and Betti tables

Let $G$ be a graph with $d$ vertices, genus $g$, and girth $b3$. Let $\overline{G}$ be the corresponding graph curve, with Betti numbers $b_{i,j}(\overline{G})$. Burnham–Rosen–Sidman–Vermeire conjecture. If $d=2g+1+p$ and $\gamma-2\leq p$, then

$$b_{\gamma-2,\gamma}(\overline{G})$$

is equal to the number of $\gamma$-cycles in $G$. This predicts that a graph's cycle structure is reflected in the Betti table of its graph curve; the paper proves a specific case for trees of cycles, while the stated general form is not resolved here.