Cubical and singular cubical homology conjecture for digital images

Let $X\subset \mathbb{Z}^n$ be a digital image with $c_1$-adjacency. The groups $\bar H_q(X)$ and $\bar H_q^{c_1}(X)$ are, respectively, the cubical and elementary cubical homology groups associated with $X$. Cubical homology conjecture. There is an isomorphism

$$\bar H_q(X)\cong \bar H_q^{c_1}(X).$$

The analogous simplicial and singular homology theories are proved isomorphic under a finiteness condition, but constructing the required cubical chain map is difficult; the cubical statement remains open.