The acyclic digraph cellular–singular cubical homology conjecture

Let $G$ be a digraph. Write $H_*^{\operatorname{cell}}(G;\mathbb R)$ for its cellular homology and $H_*^c(G;\mathbb R)$ for its singular cubical homology. The cellular–singular cubical homology conjecture. If $G$ is acyclic, then

$$H_*^{\operatorname{cell}}(G;\mathbb R)\cong H_*^c(G;\mathbb R).$$

In particular, the admissible relations among the admissible pairs are the boundary conditions in the singular cubical homology. The conjecture identifies the cellular chain construction with the singular cubical theory for acyclic digraphs; the paper motivates it through the relation between these complexes but does not state a resolution.