Let $A_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$. For any partition of $g$, one can consider the corresponding locus inside $A_g$ of products of lower-dimensional abelian varieties.

My question is: can one describe explicitly the closure of this locus in (some) compactification?

I guess part of this question is what compactification one should use. For instance, if one simply chooses a toroidal compactification $\widetilde{A}_g$, one can also extend the universal family to a family of semistable abelian varieties. Can one describe necessary and sufficient conditions for a semistable abelian variety at the boundary for it to be in the closure of such a product locus? Or perhaps if one restricts attention to rank-one degenerations, so it does not depend on a choice of toroidal compactification?

Alternatively, one could choose a point in the closure of $A_g$ inside Alexeev's space of stable semi-abelic pairs -- are there necessary and sufficient conditions in this situation?