The generic injectivity conjecture for the moduli map of self-associated sets in ${\bf P}^7$

Let $G_\omega(3,6) \subset {\bf P}^{13}$ be the isotropic Grassmannian, let $\Xi \subset G({\bf P}^7,{\bf P}^{13})$ be the open subset of linear sections meeting $G_\omega(3,6)$ transversally at $16$ distinct points, and consider the moduli map

$$\sigma: \Xi / Sp(6) \longrightarrow {\mathcal{A}}_7.$$

Generic injectivity conjecture. The map $\sigma$ is generically injective.

This conjecture concerns the moduli of self-associated sets of points in ${\bf P}^7$ that arise as linear sections of the isotropic Grassmannian. The preceding dimension count shows that the image has dimension at most $27$, whereas ${\mathcal{A}}_7$ has dimension $28$; the generic injectivity assertion describes the expected uniqueness of the corresponding section within its image.