Cohomology conjecture for the complete graph $K_5$

Let $K_5$ be the complete graph on five vertices. Complete-graph $K_5$ conjecture.

$$H^{1,3m-2}_{{\mathcal A}_m}(K_5)=\begin{cases}\mathbb{Z}_2\oplus\mathbb{Z}^{5},&m=2;\mathbb{Z}_2\oplus\mathbb{Z}_3^{4}\oplus\mathbb{Z}^{10},&m=3;\mathbb{Z}_2^{5}\oplus\mathbb{Z}_m^{10}\oplus\mathbb{Z}^{10},&m=4;\mathbb{Z}_2^{5}\oplus\mathbb{Z}_m^{10}\oplus\mathbb{Z}^{10},&m>4,\ m\text{ odd};\mathbb{Z}_2^{10}\oplus\mathbb{Z}_m^{10}\oplus\mathbb{Z}^{10},&m>5,\ m\text{ even}. \end{cases}$$

The source presents this as a conjectural computational pattern and gives no resolution.