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 $\Sigma\subset D$ 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 $K\to1$. This is a quasiconformal analog of the classical Carathéodory Theorem on the Möbius property of an injective mapping of a domain $D\subset\mathbb R^n$ which sends spheres to spheres.