Quasiconformality of the injective mappings transforming spheres to quasispheres

We prove that every injective mapping of a domain (D subset overline {{mathbb{R}^n}} ) transforming spheres Σ ? D to K-quasispheres (the images of spheres under K-quasiconformal automorphisms of (overline {{mathbb{R}^n}} )) is K′-quasiconformal with K′ depending only on K and tending to 1 as K → 1. This is a quasiconformal analog of the classical Carathéodory Theorem on the Möbius property of an injective mapping of a domain D ? Rⁿ which sends spheres to spheres.