Folha–Pacard–Zolotareva conjecture for prismatic free boundary minimal surfaces

Let $n\geq2$ 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 $\mathbb{P}_n$ denote the prismatic symmetry group, and let $\mathbb{K}_{\mathrm{crit}}$ denote the critical catenoid. Folha–Pacard–Zolotareva conjecture. For every integer $n\geq2$ there exists a free boundary minimal surface $\Xi_n^{\mathrm{FPZ}}$ in $\mathbb{B}^3$ with $n$ boundary components and genus zero, having prismatic symmetry $\mathbb{P}_n$, except for $n=2$, when it is congruent to the critical catenoid. It satisfies

$$\operatorname{area}(\Xi_n^{\mathrm{FPZ}})<2\operatorname{area}(\mathbb{B}^2),$$

and $\Xi_n^{\mathrm{FPZ}}\to\mathbb{B}^2$ with multiplicity $2$ in the sense of varifolds as $n\to\infty$. For all sufficiently large $n$, it is congruent to the genus-zero surface constructed by Folha–Pacard–Zolotareva. Its Morse index is $2n$, and its $\mathbb{P}_n$-equivariant index is $1$. The existence and convergence properties are known for sufficiently large $n$; the full family and index formula are conjectural, with the case $n=2$ having the separately known critical-catenoid index.