How close is too close for singular mean curvature flows?

Suppose $(M^i_t)_{t\in [0,T)}$, $i=1,2$, 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 $d_{H}(M^1_t,M^2_t)/(T-t)^k \rightarrow 0$. We demonstrate that $M^1_t=M^2_t$ for every $t$. This generalizes a result of Martin-Hagemayer and Sesum, who proved the case where $M^1_t$ is itself a self-similarly shrinking flow.