Calabi–Yau domain conjecture for the properly embedded one-manifold [?]

Let $\mathbb{B}$ be the unit ball and let $\Delta\subset\mathbb{B}$ be the properly embedded one-manifold given in the proof of Theorem~. Let $\overline{B}$ be a smooth compact Riemannian three-ball, let $B$ denote its interior, and let $F\colon\overline{B}\to\mathbb{B}$ be a smooth diffeomorphism. A domain is called a Calabi–Yau domain for a class of surfaces if it admits no complete, properly immersed open surface in that class with bounded mean curvature. Calabi–Yau domain conjecture. The domain

$$\mathcal{D}=B-F^{-1}(\Delta)$$

is a Calabi–Yau domain for any noncompact surface with compact boundary, possibly empty. In particular, $\mathcal{D}=\mathbb{B}-\Delta$ does not admit any complete, properly immersed open surfaces with bounded mean curvature. This conjecture proposes a broad Calabi–Yau property for domains obtained by removing the specified properly embedded one-manifold; the paper concludes with it, and no resolution is given in the supplied text.