Schoen's conjecture for limits of isoperimetric surfaces

Let $(M,g)$ be an asymptotically flat Riemannian manifold of dimension $3\leq n\leq 7$ with non-negative scalar curvature. R. Schoen has conjectured that $(M,g)$ is isometric to Euclidean space if it admits a non-compact area-minimizing hypersurface $\Sigma \subset M$. This has been proved by O. Chodosh and the first-named author in the case where $n = 3$. In this paper, we confirm this conjecture in the case where $3<n\leq 7$ and $\Sigma$ arises as the limit of isoperimetric surfaces. As a corollary, we obtain that large isoperimetric surfaces diverge unless $(M,g)$ is flat. By contrast, we show that, in dimension $3<n\leq 7$, a large part of spatial Schwarzschild is foliated by non-compact area-minimizing hypersurfaces.
Comment: All comments welcome