Nonexistence of static extensions for nonembedded locally flat boundary data

Let $(\gamma,H)$ with $H>0$ be the Bartnik boundary data of a locally flat 3-ball that is not an embedded ball in $\mathbb R^3$. Let ${\mathcal M}$ denote the space of Bartnik boundary data of immersed spheres and let ${\mathcal M}^{emb}$ denote the subspace arising from embedded spheres; let $\Pi_B$ be the boundary-data map for static vacuum extensions. Nonexistence of static extensions for nonembedded locally flat boundary data. Such boundary data lie in ${\mathcal M}\setminus{\mathcal M}^{emb}$ and are not in the image of $\Pi_B$, so they do not admit a static vacuum extension. This conjecture is open in the source; the authors also suggest an analogous statement for immersions into arbitrary static vacuum solutions, but that hedge is not recorded as a separate row.