On uniformly quasiconformal Anosov diffeomorphisms with two dimensional distributions

We prove that a transitive uniformly $u$-quasiconformal Anosov diffeomorphism with a two-dimensional unstable distribution has a globally defined stable holonomy. As a corollary, we are able to remove an additional assumption in a theorem of Kalinin-Sadovskaya, and deduce that all transitive uniformly quasiconformal Anosov diffeomorphisms are $C^{\mathrm{\infty }}$-conjugate to affine Anosov diffeomorphisms on infra-torus.