Debarre–Pareschi–Popa conjecture on minimal cohomology classes in principally polarized abelian varieties

Let $(A,\Theta)$ be an indecomposable principally polarized complex abelian variety of dimension $g$, and let $X\subset A$ be a closed subscheme of dimension $1\le d\le g-2$. Write $\theta$ for the cohomology class of $\Theta$, and call $X$ geometrically nondegenerate when it has this property in the sense of the source. A coherent sheaf is a $GV$-sheaf when its cohomological support loci have codimension at least their cohomological degree.

Debarre–Pareschi–Popa conjecture. The following are equivalent: (1) $X$ is reduced, of pure dimension, and has minimal cohomology class

$$[X]=\frac{\theta^{g-d}}{(g-d)!}.$$

(2) $X$ is a geometrically nondegenerate $GV$-subscheme, meaning that $X$ is geometrically nondegenerate and $\mathcal{I}_X(\Theta)$ is a $GV$-sheaf on $A$. (3) Either there is a smooth genus-$g$ curve $C$ and an isomorphism $(A,\Theta)\cong (JC,\Theta_C)$ identifying $X$ with $W_d(C)$, or $g=5$, $d=2$, and there is a smooth cubic threefold $Y$ and an isomorphism $(A,\Theta)\cong (JY,\Theta_Y)$ identifying $X$ with $F$, the Fano surface of lines on $Y$.