Debarre's conjecture on minimal cohomology classes

Let $(A,\Theta)$ be an irreducible principally polarised abelian variety of dimension $n$, and let $Y$ be an effective cycle on $A$ of dimension $1\leq d\leq n-2$. A cycle has minimal cohomology class when

$$[Y]=\frac{\Theta^{n-d}}{(n-d)!}.$$

Debarre's conjecture. The cycle $Y$ has minimal cohomology class if and only if $(A,\Theta)$ is the polarised Jacobian of a curve $C$ of genus $n$ and $Y$ is a translate of $W_d(C)$ or $-W_d(C)$, or $(A,\Theta)$ is the polarised intermediate Jacobian of a smooth cubic threefold and $Y$ is a translate of $F$ or $-F$.

This conjecture characterises effective cycles of minimal cohomology class on principally polarised abelian varieties. The paper proves the asserted characterisation for the intermediate Jacobian of a generic smooth cubic threefold in the relevant dimension, while the general statement is not established here.