Exterior scalar curvature comparison rigidity conjecture for convex domains

Let $\Omega$ be a convex smooth domain in $\mathbb R^n$, with $n\geq 3$, and let $(\mathbb R^n\setminus\Omega,g)$ be an asymptotically flat Riemannian manifold. Let $H_g$ and $H_0$ denote the boundary mean curvatures associated with $g$ and the Euclidean metric $g_0$, respectively. Exterior scalar curvature comparison conjecture. If

$$H_g^2g\leq H_0^2g_0,$$

then the ADM mass of $(\mathbb R^n\setminus\Omega,g)$ is non-negative, and it is zero if and only if $g$ is Euclidean. This is presented as a natural exterior generalization of Gromov's conjecture. The source does not provide a resolution of this proposed statement.