Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

We show that any integral Brakke flow coming out of a rotationally symmetric double cone with entropy at most two must stay rotationally symmetric for all time, provided the flow is smooth for a short time. We also show the existence of a non-self-similar flow coming out of a double cone with entropy at most two, and give an example of such a flow with a finite time singularity.