Generic nondegenerate singularity conjecture for mean curvature flow

Let $L_0$ be a compact embedded initial hypersurface, and let $L_t$ denote the mean curvature flow starting from $L_0$. A singularity is spherical if its model is a sphere, and cylindrical if its model is a cylinder; a cylindrical singularity is nondegenerate in the sense of Angenent–Velázquez when it satisfies the corresponding nondegeneracy condition.

Generic nondegeneracy conjecture. For a generic compact embedded initial hypersurface $L_0$, the mean curvature flow $L_t$ admits only spherical and nondegenerate cylindrical singularities.

This conjecture would extend the understanding of singularities beyond the 2-convex setting, where singularities are already known to be spherical or cylindrical and the surgery program is established. For general mean-convex hypersurfaces, higher-dimensional cylindrical factors create serious difficulties, and the conjecture remains open.