Homology conjecture for digital cycles

Let $C_m$ denote the digital cycle with $m$ vertices, and let $H_q(C_m)$ and $\bar H_q(C_m)$ be the homology theories used in the paper. Digital-cycle homology conjecture. For $m>4$,

$$H_q(C_m)=\bar H_q(C_m)= \begin{cases} \mathbb{Z} & \text{if } q=0,\\ \mathbb{Z} & \text{if } q=1,\\ 0 & \text{if } q>1. \end{cases}$$

The asserted computation is known in the even cases considered in the paper, while the odd cases, including the computation of $\bar H_2(C_5)$, remain difficult and open.