Generic multiplicity-one convergence conjecture for mean curvature flow

Let $N$ be a Riemannian manifold and let $(M(t))_{t>0}$ be a mean curvature flow starting from a generic closed hypersurface. A long-time limit has multiplicity $1$ if it occurs with multiplicity one; otherwise, the hypersurface may break into different connected components, with each component converging to a multiplicity-$1$ limit. Generic multiplicity-one convergence conjecture. Starting from a generic closed hypersurface in a Riemannian manifold, either the long-time limit has multiplicity $1$, or the hypersurface breaks into different connected components, and each one of the connected components converges to a multiplicity $1$ limit. This is motivated by the generic multiplicity-$1$ property in min-max theory. The examples in the paper show that higher-multiplicity convergence can occur, so the conjecture concerns its expected nongeneric nature.