Asymptotic growth of rank-two scattering coefficients in the Badlands

We prove that the normalized logarithmic growth rate of ordinary wall-function coefficients in every wild rank-two cluster scattering diagram exists along all integer approaches to an interior Badlands slope and is continuous in that slope. This resolves Conjecture 9.2 of Burcroff, Lee, Mou, Musiker, and Reineke.