The index and finiteness conjecture for embedded minimal surfaces in the three-sphere

Let $\gamma\in\mathbb{N}$, and let $\Sigma\subset\mathbb{S}^3$ be a closed embedded minimal surface of genus $\gamma$. Denote its Morse index by $\operatorname{Ind}(\Sigma)$, and let $\boldsymbol{\xi}_{\gamma,1}$ be the Lawson surface of genus $\gamma$. For $k_{\mathrm{excess}}\in\mathbb{N}$, consider the closed embedded minimal surfaces in the round three-sphere with index at most $2\gamma+3+k_{\mathrm{excess}}$. The index and finiteness conjecture. The index satisfies

$$\operatorname{Ind}(\Sigma)\geq 2\gamma+3,$$

with equality if and only if $\Sigma$ is congruent to the Lawson surface $\boldsymbol{\xi}_{\gamma,1}$. Moreover, for any given $k_{\mathrm{excess}}\in\mathbb{N}$, the number of such surfaces is bounded above independently of $\gamma$. This would generalize Urbano's result to arbitrary genus; the proposed lower bound is substantially stronger than the best currently known lower bound, and the conjecture is motivated by index computations for minimal surface doublings.