Lawson surfaces minimize Morse index among minimal surfaces of fixed genus

Let $M$ be a closed minimal surface in $\mathbb{S}^3$ of genus $\gamma$, and let $\operatorname{ind}(M)$ denote its Morse index. Let $\xi_{\gamma,1}$ be the Lawson surface, whose index is $2\gamma+3$. Lawson index conjecture.

$$\operatorname{ind}(M)\geq \operatorname{ind}(\xi_{\gamma,1})=2\gamma+3.$$

The value $\operatorname{ind}(\xi_{\gamma,1})=2\gamma+3$ is known, but the asserted lower bound is open.