Vacuum constraint extension conjecture for Bartnik boundary data

Let $\mathcal P$ be the space of smooth asymptotically flat Riemannian $3$-manifolds $(M,g)$ with $R_g\geq0$ and compact boundary, and let $\mathcal P_{(\gamma,H)}$ consist of those inducing $(\gamma,H)$ at the boundary. Let

$$\mathcal C_{(\gamma,H)}=\{(M,g)\in\mathcal P_{(\gamma,H)}:R_g=0\}.$$

Vacuum constraint extension conjecture. If $(M,g)$ is an extension of $(\gamma,H)$ in $\mathcal P$, then there exists an extension in $\mathcal C_{(\gamma,H)}$; equivalently,

$$\mathcal P_{(\gamma,H)}\neq\emptyset\Rightarrow\mathcal C_{(\gamma,H)}\neq\emptyset.$$

The conjecture asks whether every nonnegative-scalar-curvature extension can be replaced by a scalar-flat one with the same Bartnik boundary data. The source gives no general resolution.