Choe–Soret–Wiygul conjecture for genus-g antiprismatic surfaces

Let $g\geq2$ be an integer. A free boundary minimal surface in $\mathbb{B}^3$ is a minimal surface whose boundary lies on and meets $\partial\mathbb{B}^3$ orthogonally. Let $\mathbb{A}_{g+1}$ denote the antiprismatic symmetry group, let $\mathbb{K}_0$ be the catenoidal component appearing in the limiting configuration, and let $\Sigma_g^{\mathrm{KL}}$ be the surface in the preceding Kapouleas–Li conjecture. Choe–Soret–Wiygul conjecture. For every integer $g\geq2$ there exists a free boundary minimal surface $\Sigma_g^{\mathrm{CSW}}$ in $\mathbb{B}^3$ with $3$ boundary components, genus $g$, and antiprismatic symmetry $\mathbb{A}_{g+1}$, such that

$$\operatorname{area}(\Sigma_g^{\mathrm{KL}})<\operatorname{area}(\Sigma_g^{\mathrm{CSW}})<\operatorname{area}(\mathbb{B}^2)+2\operatorname{area}(\mathbb{K}_0),$$

and $\Sigma_g^{\mathrm{CSW}}\to\mathbb{K}_0\cup\mathbb{B}^2\cup-\mathbb{K}_0$ as $g\to\infty$ in the sense of varifolds. For all sufficiently large $g$, it coincides with the surface constructed in Theorem 1 of the source. Its Morse index is greater than $3g+6$, and its $\mathbb{A}_{g+1}$-equivariant index is $2$. The source proves the relevant symmetry and convergence for the constructed large-genus family, while existence in low genus and the asserted index information remain conjectural.