Hölder equivalence of homogeneous Moran sets

For two homogeneous Moran sets E = C([0; 1]; {n_k}; {c_k}) and E′ = C([0; 1]; {n′_k}; {c′_k}) with Hausdorff dimensions s and s′ with s′ < s such that {n_k} and {n′_k} are bounded and the spacings are uniform in some sense, we prove that there exists a homeomorphism f : E → E′ such that f is (s′/s -ϵ)-Hölder continuous but not (s′/s + ϵ)-Hölder continuous for any ϵ > 0.