Friedlander–Nadirashvili cobordism invariance conjecture

Let $M$ and $M'$ be closed manifolds of the same dimension. They are cobordant if there exists a manifold with boundary $W$ such that $\partial W=M\sqcup M'$, and $M$ is null-cobordant if there exists such a manifold with $\partial W=M$. Let $I_k(M)$ be the $k$-th Friedlander–Nadirashvili invariant, let $\dim M$ denote the dimension of $M$, and let $g_{can}$ be the canonical metric on the sphere. Cobordism invariance conjecture. The quantities $I_k$ are cobordism invariants: if $M$ is cobordant to $M'$, then

$$I_k(M)=I_k(M').$$

In particular, if $M$ is null-cobordant, then

$$I_k(M)=I_k(\mathbb{S}^{\dim M})=\Lambda_k(\mathbb{S}^{\dim M},[g_{can}]).$$

This is proposed as a higher-dimensional extension of the even-genus surface conjecture; its validity beyond the discussed surface setting remains open.