Cecil–Chi–Jensen conjecture on proper Dupin hypersurfaces with constant Lie curvatures

Let $M$ be a compact, connected proper Dupin hypersurface in $S^n$ with either four or six principal curvatures. Assume that its Lie curvatures are constant; these are the Lie-sphere invariants formed from the principal curvature data. An isoparametric hypersurface is a hypersurface with constant principal curvatures, and two hypersurfaces are Lie equivalent when they are related by a Lie sphere transformation. Cecil–Chi–Jensen conjecture. Every compact, connected proper Dupin hypersurface in $S^n$ with four or six principal curvatures and constant Lie curvatures is Lie equivalent to an isoparametric hypersurface. The original Cecil–Ryan conjecture is false because of counterexamples with nonconstant Lie curvatures. This revised conjecture remains open, although the paper notes that it is known in some cases.