The proper Dupin hypersurface conjecture in the four-curvature case

A compact embedded proper Dupin hypersurface is a compact hypersurface whose principal curvatures are constant along their corresponding curvature surfaces. In the claim below, the hypersurface lies in the unit sphere $S^5$ and has four distinct principal curvatures.

Proper Dupin hypersurface conjecture. Every compact embedded proper Dupin hypersurface in $S^5$ with four distinct principal curvatures is diffeomorphic to

$$S^1\times\mathbf{RP}^3.$$

The conjecture is motivated by the fact that the principal orbits of the isotropy representation of $\mathbf{SO}(5)/(\mathbf{SO}(2)\times\mathbf{SO}(3))$ provide examples with this diffeomorphism type. The source immediately states that it will prove the conjecture using the truth of the Poincaré conjecture in dimension $3$, so the claim is solved in the paper.