Conjecture on the Friedlander–Nadirashvili invariant of even-genus non-orientable surfaces

Let $I_k(\gamma)$ denote the $k$-th Friedlander–Nadirashvili invariant for a non-orientable surface of genus $\gamma$, and suppose that $\gamma$ is even. Even-genus conjecture. For all even $\gamma$ one has

$$I_k(\gamma)=I_k(0).$$

Moreover, the infimum defining $I_k(\gamma)$ is attained if and only if $\gamma=0$. This would determine the unresolved exact value of $I_k(\gamma)$ for even $\gamma$, beyond the bounds established in the preceding corollary.