Alexandrov immersion obstruction conjecture for the Bartnik mass

Let $\operatorname{Imm}_{A}(S^2,\mathbb R^3)$ be the space of smooth Alexandrov immersed $2$-spheres, and let $\mathcal I=\operatorname{Imm}_{A}(S^2,\mathbb R^3)\setminus\operatorname{Emb}(S^2,\mathbb R^3)$. For $F\in\mathcal I$, let $(\gamma,H)$ be the metric and mean curvature induced by the Euclidean metric. Alexandrov immersion obstruction conjecture. Either

$$\mathcal P_{(\gamma,H)}=\emptyset,$$

so that $m_B(\gamma,H)$ is undefined, or $m_B(\gamma,H)$ is not achieved by a static vacuum solution with boundary data $(\gamma,H)$. This extends the known counterexamples from embedded spheres to general non-embedded Alexandrov immersions; the source presents it as open.