Uniqueness conjecture for static asymptotically flat extensions

Let $(\Omega,g)$ be a compact Riemannian region with boundary $\partial\Omega$. Let a static asymptotically flat manifold $(S,g)$ have boundary $\Sigma$ identified with $\partial\Omega$ and satisfy the matching conditions

$$g|_{T\partial\Omega}=g|_{T\Sigma},\qquad H_{\partial\Omega}=H_{\Sigma}.$$

Uniqueness conjecture for static extensions. $(\Omega,g)$ determines a unique static asymptotically flat manifold $(S,g)$ with boundary $\Sigma\simeq\partial\Omega$ satisfying these matching conditions.

Such a uniqueness result would provide a well-defined static exterior for computing the quasi-local mass. The paper gives the matching conditions and motivates the static extension construction, but does not establish uniqueness.