Uniqueness conjecture for complete static Einstein solutions

Let $(\Sigma,g,N)$ be a complete solution of the Static Einstein equations with regular but possibly disconnected (non-empty) horizon $\partial\Sigma$. Suppose that the conformal metrics $N^2g$ and $N^{-2}g$ are complete outside given domains of compact closure on each end of $(\Sigma,g)$. Static uniqueness conjecture. The solution is either a Schwarzschild solution or a flat solution. The conjecture broadens known static black-hole uniqueness results by replacing asymptotic-flatness and end-topology assumptions with completeness of the two conformal metrics outside compact domains. Its status is not resolved in the supplied source.