Let Γ be a group which is virtually free of rank at least 2 and let Ftd(Γ) be the family of totally disconnected, locally compact groups containing Γ as a co-compact lattice. We prove that the values of the scale function with respect to groups in Ftd(Γ) evaluated on the subset Γ have only finitely many prime divisors. This can be thought of as a uniform property of the family Ftd(Γ).