Gromov's Euclidean $C^0$-rigidity conjecture

Let $g$ be a smooth complete metric on $\mathbb R^n$, where $n\geq 3$, and let $g_{\mathrm{euc}}$ denote the Euclidean metric. Assume that

$$\mathrm{scal}(g)\geq 0$$

and

$$|g-g_{\mathrm{euc}}|(x)=o(|x|^{2-n})$$

as $x\to\infty$. Gromov's Euclidean $C^0$-rigidity conjecture. Then $(\mathbb R^n,g)$ is isometric to Euclidean space. This conjecture asks whether the rigidity conclusion of the positive mass theorem follows from nonnegative scalar curvature and metric deviation from the Euclidean metric that decays faster than the Schwarzschild rate, without invoking a notion of mass. It is presented as an open question motivated by scalar-curvature stability under $C^0$ convergence.