Conjecture. If $n>1$ and $f$ is a mapping from $S^n$ to $S^n$ which maps circles into (instead of onto) circles, and whose range has n+3 distinct points any n+2 of which are in general position (in the sense of not being contained in an (n-1)-sphere), then f is a Moebius transformation.
Here, we make no any other assumption on f, e.g. continuity, injectivity, surjectivity, and so on. Circle is in the ordinary sense, i.e. round circle (or say 1-sphere), not necessarily great circle.