Cecil–Chi–Jensen revised conjecture for proper Dupin hypersurfaces

Let $M$ be a compact, connected proper Dupin hypersurface in $S^n$, with $g$ distinct principal curvatures. The hypersurface has constant Lie curvatures when its Lie curvatures are constant. Two hypersurfaces are Lie equivalent if they are mapped to one another by a Lie sphere transformation. An isoparametric hypersurface is a hypersurface in a sphere with constant principal curvatures. Cecil–Chi–Jensen's revised conjecture. If $g=4$ or $g=6$ and the Lie curvatures are constant, then $M$ is Lie equivalent to an isoparametric hypersurface. This revised conjecture was introduced to exclude the known counterexamples to the original conjecture, whose Lie curvatures are not constant. The source describes research on this revised conjecture as important, but does not establish its resolution.