Maximal-area conjecture for genus-zero free boundary minimal surfaces

Let $b\geq3$ 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. Maximal-area conjecture. For every integer $b\geq3$ there exists a genus-zero free boundary minimal surface $\Gamma_b^{\mathrm{max}}$ in $\mathbb{B}^3$ with $b$ boundary components that maximizes area among all embedded free boundary minimal surfaces in $\mathbb{B}^3$ with the same topology. It satisfies

$$\operatorname{area}(\Gamma_b^{\mathrm{max}})<\operatorname{area}(\partial\mathbb{B}^3),$$

and $\Gamma_b^{\mathrm{max}}\to\partial\mathbb{B}^3$ in the sense of varifolds as $b\to\infty$. For $b\in\{5,6,7\}$ it is congruent to $\Gamma_{b-2}^{\mathrm{CSW}}$, and for all $b\geq3$ its Morse index is $3(b-1)$. The existence and asymptotic assertions have been discussed in the literature, but the index formula and the complete conjecture remain open.