Eventual vanishing conjecture for discrete singular cubical homology of graphs

Let $G$ be a graph. Write $4\mathcal{H}_d^{\normalfont {\textrm{Cube}}}(G)4$ for its discrete singular cubical homology group in dimension $d$. Eventual vanishing conjecture. For every graph $G$, there exists an integer $N$ such that

$$\mathcal{H}_d^{\normalfont {\textrm{Cube}}}(G)=(0)$$

for all $d\geq N$.

The conjecture concerns whether the discrete singular cubical homology of every graph eventually vanishes in all sufficiently high dimensions. The paper states that it was made in earlier work and remains open, although the paper proves vanishing in every dimension $d\geq2$ for graphs containing no $3$-cycles or $4$-cycles.