Least-area conjecture for free boundary minimal surfaces in the 3-ball

Let $N_{\gamma,1}$ and $N_{\gamma,2}$ be the basic reflection surfaces and let $\Gamma_{12}$ and $\Gamma_{12}'$ be the reflection groups defined in the source. For a free boundary minimal surface $S\subset\mathbb{B}^3$, write $|S|$ for its area and $\chi(S)$ for its Euler characteristic. Free-boundary least-area conjecture.

$$|S|\geq\frac12\Sigma_1(N_{\gamma,1},\Gamma_{12})\quad\text{when }\chi(S)\text{ is odd and }\gamma=\frac{1-\chi(S)}2$$

and

$$|S|\geq\frac12\Sigma_1(N_{\gamma,2},\Gamma_{12}')\quad\text{when }\chi(S)\text{ is even and }\gamma=\frac{-\chi(S)}2.$$

Here $\Sigma_1$ is the corresponding normalized first Steklov eigenvalue. The conjecture asserts that the constructed families have least area among free boundary minimal surfaces with the prescribed Euler characteristic, while accounting for the larger reflection group needed when the Euler characteristic is even.