The generic-fiber conjecture for two-body eigenvector varieties

Let $\mathcal{W}_{k,m}$ denote the space of two-body operators, viewed as a subspace of $\operatorname{Sym}^2(\wedge^k \mathbb{C}^m)$, and let $\mathcal{E}(\mathcal{W}_{k,m})$ be its eigenvector variety. Let $\mathcal{J}(\mathcal{W}_{k,m})$ denote the corresponding horizontal incidence variety, with projection

$$\mathcal{J}(\mathcal{W}_{k,m}) \longrightarrow \mathcal{E}(\mathcal{W}_{k,m}).$$

Generic-fiber conjecture. Equality holds in the dimension bound

$$\dim \mathcal{E}(\mathcal{W}_{k,m})=\min\left(\binom{m}{k}-1,\binom{\binom{m}{2}+1}{2}-2\right).$$

Equivalently, the generic fiber of the projection $\mathcal{J}(\mathcal{W}_{k,m})\to\mathcal{E}(\mathcal{W}_{k,m})$ is one-dimensional, spanned by the identity direction in $\mathcal{W}_{k,m}$. The conjecture specifies the expected dimension of the two-body eigenvector variety and the minimal generic fiber forced by the identity operator; its resolution is not established in the supplied text.