Maximal static vacuum immersion conjecture

Let $(\gamma,H)$ be Bartnik boundary data on $S^2$, normalized by $\operatorname{area}_{\gamma}(S^2)=1$. A maximal static vacuum solution is an asymptotically flat static vacuum triple $(M,g,u)$ with $u>0$ that admits no proper smooth extension as a static vacuum solution. Maximal static vacuum immersion conjecture. There is a constant $H_1>0$ such that, if

$$H\geq H_1>0,$$

then there is a maximal static vacuum solution $(M,g,u)$ and an immersion $F:S^2\rightarrow M$ realizing $(\gamma,H)$, meaning that the induced boundary data on $F(S^2)$ are $(\gamma,H)$. The conjecture is a more reasonable replacement for the unrestricted maximal static-vacuum immersion problem, which is incompatible with earlier obstructions; its general validity is open.