The conformal conjecture for asymptotically flat manifolds with boundary

Let $(M^n,g)$, $n \ge 3$, be a complete, asymptotically flat, smooth $n$-manifold with nonempty compact boundary $\Sigma=\partial M$ smooth with respect to both the smooth structure and the metric $g$. Conformal conjecture. There exists a positive harmonic function $u(x)$ on $(M^n,g)$ tending to one at infinity such that, after setting

$$\overline{g}=u(x)^{4/(n-2)}g,$$

and letting $\widetilde{\Sigma}$ be the outermost minimal enclosure of $\Sigma$ in $(M^n,\overline{g})$, the hypersurfaces $\widetilde{\Sigma}$ and $\Sigma$ have the same $(n-1)$-volume in $(M^n,\overline{g})$, and $\widetilde{\Sigma}$ has zero mean curvature, at least in some weak sense compatible with the Riemannian Penrose inequality. When true, this conjecture permits application of the Riemannian Penrose inequality to asymptotically flat manifolds with nonnegative scalar curvature and compact boundary; its status is not established in the supplied text.