Removing a technical boundary assumption in the noncollapsing result via elliptic regularization

Show that Item (iii) (noncollapsing) of Proposition \ref{Prop:MCF with boundary} holds without the technical assumption \ref{Item_EllReg_BdyFamilyExt}, which postulates the existence of a smooth monotone boundary deformation (Γ_t) on ∂B_{r_0}∩Ω connecting M_t∩∂B_{r_0} for t∈[0,T] to (S^{n−k}(r_0′)×R^k)∩∂B_{r_0} for t≥T+1.

Background

Proposition \ref{Prop:MCF with boundary} establishes mean convexity, uniqueness, and noncollapsing of the level set flow constructed by elliptic regularization under several assumptions, including a boundary deformation hypothesis (Item \ref{Item_EllReg_BdyFamilyExt}).

The authors note this hypothesis is only used to prove the noncollapsing part and conjecture it can be removed, which would strengthen the robustness of the noncollapsing conclusion without extra boundary data.