The Conformal Conjecture for asymptotically flat three-manifolds with boundary

Let $(N,h)$ be a smooth asymptotically flat 3-manifold with compact, smooth, nonempty boundary $\Sigma$, with $h$ extending smoothly to $\Sigma$. Given $\epsilon>0$, there exists a smooth, positive function $u$ on $N$ and metric $h_0=u^4h$. Let $\widetilde \Sigma$ be the outermost minimal area enclosure of $\Sigma$ with respect to $h_0$.

Conformal Conjecture. The function $u$ is harmonic with respect to $h$ and tends to 1 at infinity; in the metric $h_0$, the areas of $\widetilde \Sigma$ and $\Sigma$ differ by at most $\epsilon$; and $\widetilde \Sigma$ is disjoint from $\Sigma$.

The third condition makes $\widetilde \Sigma$ a smooth zero-mean-curvature surface in $h_0$. The conjecture is known in the spherically symmetric case and is needed for the arguments establishing the ZAS inequality.