Existence conjecture for complete free boundary minimal planes

Let $M=(\mathbb R_{+}^{3},g)$ be an asymptotically flat three-dimensional Riemannian manifold with boundary, and let $p\in\partial M$. A free boundary minimal plane is a complete, non-compact, properly embedded minimal plane in $M$ whose boundary lies in $\partial M$ and meets $\partial M$ orthogonally. Existence conjecture. If $M$ contains no compact free boundary minimal surface, then there exists a complete non-compact free boundary minimal plane passing through $p$. This conjecture is motivated by analogous existence results for complete non-compact minimal planes in asymptotically flat three-dimensional manifolds without boundary; the boundary case is posed here as an open problem.