Failure of Bartnik mass minimization for nonembedded locally flat balls

Let $F:\bar B\rightarrow\mathbb R^3$ be a smooth immersion of a 3-ball that is not an embedding, and equip the ball with the pulled-back Euclidean metric $F^*(g_{\mathrm{Eucl}})$. Failure of Bartnik mass minimization for nonembedded locally flat balls. The region $(\bar B,F^*(g_{\mathrm{Eucl}}))$ admits no admissible extension realizing its Bartnik mass. The paper notes that the corresponding stronger version of the Bartnik mass conjecture is disproved for a particular locally flat ball; this general formulation is presented as a conjecture and its status is not established in the source.