How close is too close for singular mean curvature flows?

Suppose (Mti)t∈[0,T), i=1,2, are two mean curvature flows in Rn+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, dH, satisfies dH(Mt1,Mt2)/(T-t)k→0. We demonstrate that Mt1=Mt2 for every t. This generalizes a result of Martin-Hagemayer and Sesum, who proved the case where Mt1 is itself a self-similarly shrinking flow.