Conjecture on codimension-one singular Riemannian foliations for non-negatively curved manifolds

Let $M$ be a closed simply connected Riemannian manifold with non-negative sectional curvature. A singular Riemannian foliation of codimension 1 is a singular Riemannian foliation whose leaves have codimension one on the regular stratum. A bundle-like metric is a Riemannian metric adapted to the foliation. Codimension-one foliation conjecture. Every such $M$ admits a singular Riemannian foliation of codimension 1, either for the given metric or for a different bundle-like metric.

Such a foliation would imply that $M$ admits a splitting into two disk bundles. The conjecture is essentially attributed to Karsten Grove, who conjectured that every compact non-negatively curved manifold splits as a union of two disk bundles; it remains open even up to homeomorphism.