Factorial denominators of rank-two scattering coefficients

We prove that, for every pair of positive integers $i,j$, the universal rank-two cluster scattering coefficient $\tau(i,j)$ satisfies $\max(i,j)!\,\tau(i,j)\in\mathbb{Z}[b,c,g]$. This resolves the factorial-denominator conjecture of Elgin, Reading, and Stella (Conjecture 4).