Bernoulli foliation conjecture for asymptotically flat 3-manifolds

Let $(M,g)$ be an asymptotically flat 3-manifold with nonnegative scalar curvature and positive ADM mass. A Bernoulli surface is a surface $S=\partial A$ for a compact set $A$ such that there exists a harmonic function on $M\setminus A$ approaching $1$ at infinity, equal to zero on $S$, and having constant normal derivative on $S$. Bernoulli foliation conjecture. There exists a compact set $K$ such that $M\setminus K$ is foliated by Bernoulli surfaces of spherical topology.

This conjecture is motivated by the canonical foliation of the asymptotic region by stable constant-mean-curvature surfaces. Results on Bernoulli foliations relative to a fixed surface are known, but the conjectured foliation relative to infinity remains open.