The avoidance principle for mean curvature flow says that, given two disjoint compact, embedded hypersurfaces, when we run the mean curvature flow, they will remain disjoint.

In fact it is known that we can relax the assumptions to “one is compact and embedded, the other one is just embedded”.

Can we relax the assumptions even more? e.g. to “two complete, embedded hypersurfaces”, or adding "properly embeddedness"? Or is there any known counterexample preventing us from doing so?