Lawson surfaces minimize area among minimal surfaces of fixed genus

Let $\mathcal{A}_\gamma$ be the minimum area among all minimal surfaces in $\mathbb{S}^3$ of genus $\gamma$, and let $\xi_{\gamma,1}$ be the Lawson surface of genus $\gamma$. Lawson area-minimization conjecture. The minimum area $\mathcal{A}_\gamma$ is uniquely attained by $\xi_{\gamma,1}$. This is presented as a special case of a conjecture characterizing Lawson surfaces as Willmore minimizers.