Cohomology conjecture for the square with a diagonal

Let $P_3|P_3$ denote the graph formed by gluing two triangles along an edge, and let $H^{i,j}_{{\mathcal A}_m}(G)$ denote the corresponding graph cohomology. Square-with-diagonal conjecture. For $m>2$,

$$H^{1,2m-1}_{{\mathcal A}_m}(P_3|P_3)=\mathbb{Z}_m^2\oplus\mathbb{Z}.$$

The paper reports verification for $m\leq 15$, but does not establish the general claim.