The minimal-element area and genus growth conjecture for CMC surfaces

Let $M\subset \mathbb{R}^3$ satisfy the hypotheses of the Minimal Element Theorem, and let $\Sigma\in\mathcal{T}(M)$ be a minimal element. For $n\in\{1,2,3\}$, consider the area and genus of $\Sigma$ inside the Euclidean ball $\mathbb{B}(R)$. Minimal-element growth conjecture. The limits

$$\lim_{R\to\infty}\operatorname{Area}[\Sigma\cap\mathbb{B}(R)]R^{-n}\quad\text{and}\quad \lim_{R\to\infty}\operatorname{Genus}[\Sigma\cap\mathbb{B}(R)]R^{-n}$$

exist, possibly with value $+\infty$, and

$$A_{\inf}(\Sigma,n)=A_{\sup}(\Sigma,n)=\lim_{R\to\infty}\operatorname{Area}[\Sigma\cap\mathbb{B}(R)]R^{-n},$$ $$G_{\inf}(\Sigma,n)=G_{\sup}(\Sigma,n)=\lim_{R\to\infty}\operatorname{Genus}[\Sigma\cap\mathbb{B}(R)]R^{-n}.$$

The conjecture concerns the asymptotic area and genus growth of minimal elements in the translation space of a complete, noncompact, connected, embedded constant-mean-curvature surface. The source notes that the corresponding assertion holds for $n=1$ by an item of the Minimal Element Theorem; the cases $n=2$ and $n=3$ are not resolved in the supplied text.