Stoïlow factorization for quasiregular mappings in all dimensions

We generalize to higher dimensions the classical Stoïlow factorisation theorem; we show that any quasiregular mapping $f$ of the Riemann $n$-sphere $\hat{\mathbb{R}}^n \approx \mathbb{S}^n$ can be written in the form $f=\varphi \circ h$, where $h:\mathbb{S}^n \to \mathbb{S}^n$ is quasiconformal and $\varphi$ is a uniformly quasiregular mapping, hence rational with respect to some bounded measurable conformal structure.