The homology conjecture for total 3-cut complexes of complete graph–cycle products

Let $K_m$ be the complete graph on $m$ vertices, let $C_n$ be the cycle graph on $n$ vertices, and let $\Delta_{3}^t(K_m\square C_n)$ denote their total $3$-cut complex. The total 3-cut cycle-product homology conjecture.

For $m\geq3$ and $n=3$,

$$\widetilde{H}_i\bigl(\Delta_{3}^t(K_m\square C_n)\bigr)= \begin{cases} \mathbb{Z}^{\frac{1}{2}(m^2-3m+2)},&\text{if }i=mn-6,\\ 0,&\text{otherwise}. \end{cases}$$

For $m\geq2$ and $n\in\{4,5\}$,

$$\widetilde{H}_i\bigl(\Delta_{3}^t(K_m\square C_n)\bigr)= \begin{cases} \mathbb{Z}^{\frac{m-1}{2}((m-1)n^2-(m-3)n-2)},&\text{if }i=mn-6,\\ 0,&\text{otherwise}. \end{cases}$$

For $m\geq2$ and $n\geq6$,

$$\widetilde{H}_i\bigl(\Delta_{3}^t(K_m\square C_n)\bigr)= \begin{cases} \mathbb{Z}^{\frac{1}{2}((m-1)^2n^2-(m^2-4m+3)n+2)},&\text{if }i=mn-6,\\ 0,&\text{otherwise}. \end{cases}$$

The conjecture is based on computed examples for total $3$-cut complexes of complete graph–cycle products; the source does not provide a resolution.