Ancient mean curvature flow asymptotic to Simons cone

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to $O(n)\times O(n)$ symmetric Simons cone for $n \geq 5$, and lies on one side of the cone has to have a unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation.