Pareschi–Popa conjecture on summands of theta divisors

Let $(A,\Theta)\in {\mathscr A}_g({\mathbb C})$ be an indecomposable principally polarized abelian variety. A nontrivial sum decomposition of its theta divisor is an expression $\Theta=X+Y$ with $X,Y\subset A$ irreducible subvarieties of positive dimension such that the addition morphism $X\times Y\twoheadrightarrow\Theta$ is generically finite.

Pareschi–Popa conjecture. If $\Theta=X+Y$ decomposes nontrivially as a sum, then either $(A,\Theta)$ is the Jacobian of a smooth projective curve, or $(A,\Theta)$ is the intermediate Jacobian of a smooth cubic threefold, with $g=5$.

This conjecture characterizes the Jacobian locus, together with the intermediate Jacobian of a smooth cubic threefold, through decompositions of theta divisors. It was reformulated by Schreieder, who proved it for curve summands; the general statement remains open.