Bryant's conjecture for complete minimal hypersurfaces in the 4-sphere

Let $M$ be a complete minimal hypersurface of constant scalar curvature in $\mathbb{S}^4(1)$.

Bryant's conjecture. $M$ is isoparametric.

This is the noncompact analogue of the classification of closed minimal hypersurfaces in $\mathbb{S}^4(1)$. The compact case is stated to be completely classified, whereas the complete case without compactness remains open.