Debarre–Iliev–Manivel conjecture on the period fiber of ordinary Gushel–Mukai threefolds

Let $\mathcal{X}_{10}$ be the moduli space of ordinary Gushel–Mukai threefolds, let $\mathcal{A}_{10}$ be the moduli space of $10$-dimensional principally polarised abelian varieties, and let

$$\mathcal{P}:\mathcal{X}_{10}\rightarrow\mathcal{A}_{10}$$

be the classical period map. For an ordinary GM threefold $X$, let $J(X)$ be its intermediate Jacobian, let $\mathcal{C}_m(X)$ be the relevant surface of conics, and let $M_G(2,1,5)$ be the corresponding moduli surface. Debarre–Iliev–Manivel conjecture. A general fiber $\mathcal{P}^{-1}([J(X)])$ through $X$ is the union of $\mathcal{C}_m(X)/\iota$ and a surface birationally equivalent to $M_G(2,1,5)/\iota'$, where $\iota$ and $\iota'$ are geometrically meaningful involutions. This describes the expected geometry of the fibers of the classical period map; the conjecture is cited as an earlier prediction and is not resolved by the general results stated here.