Complete proper minimal surfaces in convex bodies of $\mathbb R^3$, II. The behavior of the limit set

Let $D$ be a regular, strictly convex bounded domain of $\mathbb{R}^3$, and consider a Jordan curve $\Gamma \subset \partial D$. Then, for each $\varepsilon>0$, we obtain the existence of a complete proper minimal immersion $\psi_\varepsilon \colon \mathbb{D} \rightarrow D$ satisfying that the Hausdorff distance $\delta^H(\psi_\varepsilon(\partial \mathbb{D}), \Gamma) < \varepsilon,$ where $\psi_\varepsilon(\partial \mathbb{D})$ represents the limit set of the minimal disk $\psi_\varepsilon(\mathbb{D})$.