Finiteness conjecture for components of the singular set of mean curvature flow

Let $M_t\subset\mathbb{R}^{n+1}$ be a mean curvature flow with only cylindrical singularities, starting at a closed smooth embedded hypersurface. The space-time singular set is conjectured to have only finitely many components.

Finiteness conjecture. The space-time singular set has only finitely many components.

The preceding discussion recalls that dimension-reduction arguments give an upper bound of $n-1$ for the dimension of the singular set of a mean curvature flow with only cylindrical singularities. This conjecture asks for the stronger topological conclusion that the singular set has finitely many components.