Vanishing of cubical homology for graphs without triangles or quadrilaterals

Let $G$ be an undirected graph, and let $\mathcal{H}_n^{\normalfont {\textrm{Cube}}}(G)$ denote its $n$-th cubical homology group. Suppose that $G$ contains no $3$-cycles and no $4$-cycles. Cubical vanishing conjecture. Then

$$\mathcal{H}_n^{\normalfont {\textrm{Cube}}}(G) \cong (0)$$

for $n\ge 2$. The analogous statement for path homology was proved immediately beforehand; the cubical version is supported only by limited computational evidence and remains open.