Assume the statement fails, then there is a constant $\tau>0$, a sequence $\epsilon_{i}\rightarrow 0$, and maps $\phi_{i}:\hat{A}(30,80)\rightarrow\mathbb{R}^{4}$ with $|\phi_{i}^{*}g_{0}-g_{0}|_{C^{4}(\hat{A}(30,80))}\leq\epsilon_{i}$, but for any isometry $P$ we have $|P\circ\phi_{i}-Id|_{C^{3}(\hat{A}(40,70))}\geq\tau$. Then $\phi_{i}$ converges to a map $\phi_{\infty}$ in $C^{3}(\hat{A}(40,70))$, such that $\phi_{\infty}^{*}g_{0}=g_{0}$. So $\phi_{\infty}$ is an isometry of $\mathbb{R}^{4}$. Since $|\phi_{\infty}^{-1}\circ\phi_{i}-Id|_{C^{3}(\hat{A}(40,70))}$ converges to zero as $i$ goes to infinity. We arrive at a contradiction. ∎