Grove's conjecture on DDBD structures

Let $M$ be a closed simply connected manifold admitting a Riemannian metric of nonnegative sectional curvature. Grove's conjecture. Then $M$ has a DDBD structure, that is, an isoparametric foliation under some Riemannian metric. This is proposed as a compact analogue of the soul theorem: the source notes that it would have consequences for the differential classification of closed simply connected four-manifolds with nonnegative sectional curvature, but no general proof is known.