Kapouleas–Li conjecture for antiprismatic free boundary minimal 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, and let $\mathbb{K}_{\mathrm{crit}}$ denote the critical catenoid. Kapouleas–Li conjecture. For every integer $g\geq2$ there exists a free boundary minimal surface $\Sigma_g^{\mathrm{KL}}$ in $\mathbb{B}^3$ with $3$ boundary components, genus $g$, and antiprismatic symmetry $\mathbb{A}_{g+1}$, satisfying

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

with $\Sigma_g^{\mathrm{KL}}\to\mathbb{B}^2\cup\mathbb{K}_{\mathrm{crit}}$ as $g\to\infty$ in the sense of varifolds. For all sufficiently large $g$, it is congruent to both the surface constructed by Kapouleas–Li and the surface $\Sigma_g^{\mathrm{Ket}}$ from Ketover. Its Morse index is $2g+6$, and its $\mathbb{A}_{g+1}$-equivariant index is $1$. The conjecture extends known large-genus constructions and predicts their precise congruence and index; the low-genus existence and the asserted index formula remain conjectural.