Dupin hypersurfaces with constant Laguerre curvatures

We consider proper Dupin hypersurfaces of the Euclidean space Rn+1, that admit principal coordinate systems and have n distinct nonvanishing principal curvatures. We obtain explicitly all such hypersurfaces that have constant Laguerre curvatures. In particular, we show that they are determined by n-2 Laguerre curvatures and two other constants, one of them being nonzero.