White's generic regularity conjecture for area-minimizing cycles

Let $M$ be a Riemannian manifold and let a homologically area-minimizing cycle in $M$ have a singular set. White's generic regularity conjecture. Singularities in a homologically area-minimizing cycle in a Riemannian manifold should disappear after a generic perturbation of the metric. This question concerns whether generic metrics eliminate singularities from area-minimizing representatives when smooth representatives are not excluded by topology. It is known for embedded geodesics and for $2$-dimensional area-minimizing submanifolds of arbitrary codimension, while the general question remains open.