Branch sets of uniformly quasiregular maps

Let $n\geq 2$ and $f:{\Bbb S}^n\to {\Bbb S}^n$ be a quasiregular mapping with branch set $B_f$, the set where $f$ fails to be locally injective. We show that there is a quasiregular mapping $g:{\Bbb S}^n\to {\Bbb S}^n$ with $B_g = B_f$ and such that $g$ can be chosen to be conformal (rational) with respect to some measurable Riemannian structure on ${\Bbb S}^n$. Hence $g$ is uniformly quasiregular. That is, $g$ and all its iterates are quasiregular with a uniform bound on the dilatation.