Non-surjectivity of the static vacuum boundary map near Schwarzschild data

Let $\gamma\in\operatorname{Met}^{m,\alpha}(S^2)$ have non-constant Gauss curvature, and let $C_+^{m-1,\alpha}(S^2)$ denote the positive mean-curvature data. Non-surjectivity of the static vacuum boundary map near Schwarzschild data. There is a neighborhood $U_\gamma\subset C_+^{m-1,\alpha}(S^2)$ with $0\in\overline{U_\gamma}$ such that, for every $H\in U_\gamma$, the boundary data $(\gamma,H)$ do not bound a static vacuum metric $(M,g,u)$. In particular, $\Pi_B$ is not surjective near the Schwarzschild metric. The source marks this as open; it describes a second region where the preceding conjecture breaks down and asks whether such extensions exist when the boundary is not outer-minimizing.