How close is too close for singular mean curvature flows?

Suppose (Mⁱₜ)t∈ [0,T) ( M t i ) t ∈ [ 0 , T ) , $$i=1,2$$ i = 1 , 2 , are two mean curvature flows in Rⁿ⁺¹ 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, dH d H , satisfies dH(M¹ₜ,M²ₜ)/(T-t)ᵏ → 0 d H ( M t 1 , M t 2 ) / ( T - t ) k → 0 . We demonstrate that M¹ₜ=M²ₜ M t 1 = M t 2 for every t . This generalizes a result of Martin-Hagemayer and Sesum, who proved the case where M¹ₜ M t 1 is itself a self-similarly shrinking flow.