Palais–Terng conjecture on integrable orbit-normal distributions

Let a compact Lie group $G$ act isometrically on a Riemannian manifold $M$, and suppose that the distribution of normal spaces to the regular $G$-orbits is integrable. Palais–Terng conjecture. There exists a complete, totally geodesic immersed section for the action of $G$ that intersects every orbit perpendicularly. This would characterize a broad class of actions with integrable normal distributions as admitting global sections, extending the structure theory of polar actions. The source gives no resolution of the conjecture.