Surjective sphere-preserving maps of the extended Euclidean space

Let $n$ and $r$ be the dimensions under consideration, and let $f:\hat{\mathbb{R}}^n\to\hat{\mathbb{R}}^n$ be a map that sends every $r$-dimensional sphere into an $r$-dimensional sphere. Sphere-preserving Möbius conjecture. If $f$ is surjective, then $f$ is a Möbius transformation. This conjecture was posed in the context of removing injectivity assumptions from sphere-preserving characterizations; the corresponding Euclidean hyperplane statement is known, while this spherical assertion is presented as open here.