Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation

In this paper, we construct a family of mean curvature flow which converges to an area minimizing, strictly stable hypercone $\text{mC}$ after type I rescaling, and converges to the Hardt-Simon foliation of the cone after a type II rescaling provided the cone satisfies some technique conditions. The difference from Velázquez’s previous results is that we drop the symmetry condition on the cone.