Cecil–Ryan conjecture on compact proper Dupin hypersurfaces

Let $M$ be a compact, connected proper Dupin hypersurface in $S^n$ or ${\bf R}^n$. Cecil–Ryan conjecture. $M$ is Lie equivalent to an isoparametric hypersurface in $S^n$. The conjecture was proposed when all known examples were Lie equivalent to isoparametric hypersurfaces, but it was later disproved by counterexamples of Miyaoka and Ozawa.