Schoen's conjecture for limits of isoperimetric surfaces

Let $(M,g)$ be an $n$-dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature that admits a noncompact area-minimizing hypersurface $\Sigma \subset M$. In the case where $n = 3$, O. Chodosh and the first-named author have proven that $(M, g)$ is necessarily isometric to Euclidean space, confirming a conjecture of R. Schoen. In this paper, we extend this result to dimension $3 < n \leq 7$ provided that $\Sigma$ arises as a limit of isoperimetric surfaces. By contrast, we prove that when $3 < n \leq 7$, there is no such result for general noncompact area-minimizing $\Sigma \subset M$, even when additional assumptions on the stability of $\Sigma$ are imposed.