Local flatness conjecture for the vacuum Bartnik mass

Let $\Omega$ be a compact domain with boundary data induced by a nonnegative-scalar-curvature metric, and let

$$m_B^0(\Omega)=\inf\{m_{ADM}(M,g):(M,g)\in\mathcal C_{(\gamma,H)}\},$$

where $\mathcal C_{(\gamma,H)}$ is the set of asymptotically flat extensions with scalar curvature zero and boundary data $(\gamma,H)$. Local flatness conjecture. One has

$$m_B^0(\Omega)=0,$$

only if $\Omega$ is locally flat. This is a rigidity question for the modified vacuum Bartnik mass; the positive mass theorem gives nonnegativity, while the stronger strict-positivity assertion is what remains at issue.