Bridge-count conjecture for graph curves

Let $G$ be a graph and let $\overline{G}\subset \mathbb{P}^{n}$ be its graph curve. Write $b_{i,j}(\overline{G})$ for the graded Betti numbers, and call an edge of $G$ a bridge if removing it disconnects $G$. Bridge-count conjecture. The Betti number

$$b_{n-1,n}(\overline{G})$$

is equal to the number of bridges in $G$. This connects a graph-theoretic connectivity invariant with the Betti table of the associated graph curve; the paper proves the assertion for trees of cycles, but the general statement is presented as a conjecture.