How close is too close for singular mean curvature flows?

Suppose $(M_t^i{)}_{t\in [0,T)}$, $i=1,2$, are two mean curvature flows in ${\mathbb{R}}^{n+1}$ encountering a multiplicity one compact singularity at time $T$, in such a manner that for every $k$, the Hausdorff distance between the two flows, $d_H$, satisfies $d_H(M_t^1,M_t^2)\mathrm{/}(T-t{)}^k\to 0$. We demonstrate that $M_t^1=M_t^2$ for every $t$. This generalizes a result of Martin-Hagemayer and Sesum, who proved the case where $M_t^1$ is itself a self-similarly shrinking flow.