Friedlander–Nadirashvili invariant conjecture for closed surfaces

Let $\Sigma$ be a closed surface, and let $I(\Sigma)$ denote its Friedlander–Nadirashvili invariant, defined as the infimum over conformal classes $C$ on $\Sigma$ of the supremum $\nu(C)$ of the first eigenvalue among area-one metrics in $C$. Friedlander–Nadirashvili conjecture. For any closed surface $\Sigma$ other than the projective plane $\mathbb{R}P^2$, one has

$$I(\Sigma)=8\pi.$$

The invariant is already known to equal $8\pi$ for the sphere, torus, and Klein bottle, while the conjecture concerns all remaining closed surfaces except $\mathbb{R}P^2$.