The existence conjecture for codimension-one singular Riemannian foliations

A singular Riemannian foliation on a Riemannian manifold is a partition into connected immersed submanifolds such that smooth vector fields tangent to the leaves span the tangent spaces of the leaves, and geodesics perpendicular to one leaf remain perpendicular to every leaf they meet. A foliation has codimension one when its leaves have codimension one, and it is closed when its leaves are closed subsets. A Riemannian manifold has positive (respectively, non-negative) sectional curvature when all sectional curvatures are positive (respectively, non-negative).

The existence conjecture. Every simply connected Riemannian manifold $(M,\mathtt{g})$ with positive (respectively, non-negative) sectional curvature admits a codimension-one singular Riemannian foliation $\mathcal{F}$, possibly with respect to a different Riemannian metric.

This conjecture predicts a broad relationship between positive curvature and the topology of singular foliations. The supplied text does not state whether it is known or open, so its resolution status requires verification.