Dupin hypersurfaces with four principal curvatures, II

If $M$ is an isoparametric hypersurface in a sphere $S^n$ with four distrinct principal curvatures, then the principal curvatures $κ_1,...,κ_4$ can be ordered so that their multiplicities satisfy $m_1=m_2$ and $m_3=m_4$, and the cross-ratio $r$ of the principal curvatures (the Lie curvature) equals -1. In this paper, we prove that if $M$ is an irreducible connected proper Dupin hypersurface in $\R^n$ (or $S^n$) with four distinct principal curvatures with multiplicities $m_1=m_2 \geq 1$ and $m_3=m_4=1$, and constant Lie curvature $r=-1$, then $M$ is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and $r$ is merely assumed to be constant.