Existence of a Bartnik minimal mass extension via intrinsic-flat compactness

Let $\Omega_0$ be a region and let $\mathcal{PM}(\Omega_0)$ denote the class of asymptotically flat extensions containing an isometric image of $\Omega_0$. Bartnik extension compactness conjecture. There exists a sequence

$$M_j\in\mathcal{PM}(\Omega_0),\qquad m_{ADM}(M_j)\to m_B(\Omega_0),$$

with a limit $M_j\to M_\infty$ such that

$$m_{ADM}(M_\infty)=m_B(\Omega_0)=\lim_{j\to\infty}m_{ADM}(M_j),$$

where $M_\infty$ is asymptotically flat, smooth, scalar flat, and static. The proposed approach addresses existence of the minimal mass extension, but the required compactness and asymptotic-flatness control remain open.