Mean curvature flow with generic low-entropy initial data II

We prove that the mean curvature flow of a generic closed embedded hypersurface in ${\mathbb{R}}^4$ or ${\mathbb{R}}^5$ with entropy $\le 2$, or with entropy $\le \lambda ({\mathbb{S}}^1)$ if in ${\mathbb{R}}^6$, encounters only generic singularities.