Covering conjecture for discrete singular cubical homology of graphs

Let $G$ be a graph, and let ${\mathcal K}=\{K_i\}$ be a covering of $G$ by subgraphs such that every edge, $3$-cycle, and $4$-cycle of $G$ is contained in some $K_i$. Let $\mathcal{C}^{\mathcal K}(G)$ denote the cubical chain complex generated by cubes whose images are contained in some $K_i$. Covering conjecture. One should have

$$\mathcal{H}^{\normalfont {\textrm{Cube}}}(G)\approx\mathcal{H}^{\normalfont {\textrm{Cube}}}(\mathcal{C}^{\mathcal K}(G)).$$

The conjecture proposes a Mayer–Vietoris-type covering principle for discrete singular cubical homology. The source presents it as a speculation and does not establish it; the paper notes that related covering results are available for graphs with no $3$-cycles or $4$-cycles, where the higher-dimensional homology groups vanish.