The generic regularity hypothesis for least-area hypersurfaces

Let $M^{n+1}$ be a closed smooth manifold, let $H_n(M;\mathbb{Z})$ denote its codimension-one homology group, and let a representative of a homology class mean a smooth hypersurface in that class. Generic regularity hypothesis. There is a Baire generic set of metrics $\mathcal{G}$ such that, for every $g\in\mathcal{G}$ and every $\sigma\in H_n(M;\mathbb{Z})$, there is a smooth representative $\Sigma\in\sigma$ of least area. This hypothesis would address the singularities of area-minimizing hypersurfaces in high dimensions and provide a route to applying minimal-surface methods more broadly. The source says the conjecture has appeared in various forms since at least the 1980s and gives no resolution evidence.