Cohomology conjecture for the complete graph $K_4$

Let $K_4$ be the complete graph on four vertices. Complete-graph $K_4$ conjecture. For every $m\geq4$,

$$H^{1,2m-1}_{{\mathcal A}_m}(K_4)=\mathbb{Z}_m^4\oplus\mathbb{Z}_2\oplus\mathbb{Z}^2.$$

The paper reports verification for $m\leq14$, but does not establish the stated range in general.