For $\epsilon>0$ sufficiently small, there is a constant $K(\epsilon)>0$ which goes to zero as $\epsilon$ tends to zero, such that for any smooth map $\phi:\hat{A}(30,80)\rightarrow\mathbb{R}^{4}$ with $|\phi^{*}g_{0}-g_{0}|_{C^{4}(\hat{A}(30,80))}\leq\epsilon$, there is an isometry $P$ of $\mathbb{R}^{4}$ such that $|P\circ\phi-Id|_{C^{3}(\hat{A}(40,70))}\leq K(\epsilon)$.