Rigidity in the Positive Mass Theorem with $C^0$ Decay

Let $g$ be a smooth metric on ${\mathbb{R}}^3$ with non-negative scalar curvature. We show that if $g$ satisfies $\mathrm{\mid }g(x)-g_{\text{euc}}(x)\mathrm{\mid }=O(\mathrm{\mid }x{\mathrm{\mid }}^{-1-\tau })$ for some $\tau >0$ then $g$ must be flat.