Colorability conjecture for $G_\alpha(m;1,1,c)$ with nontrivial gcd

Let $G_\alpha(m;1,1,c)$ be the alpha family, and suppose $\gcd(c,m)>1$. Nontrivial-gcd colorability conjecture. The graph $G_\alpha(m;1,1,c)$ is 3-edge-colorable. The paper notes that the case is more complicated because the loop edges form $\gcd(c,m)$ disjoint cycles, and gives no general proof.