The Relative Manin-Mumford conjecture for semi-abelian schemes

Let $\mathcal{G}/S$ be a semi-abelian scheme of relative dimension $g+n$, with toric rank $n$. Let $Z$ be an irreducible closed subvariety of $\mathcal{G}$. If $Z(k)\cap\mathcal{G}_{tor}$ is Zariski dense in $Z$, then either $Z$ is contained in a proper special subvariety of $\mathcal{G}$, or $\dim Z\geq g+n$.

Relative Manin-Mumford conjecture. The stated alternative holds.

This is a reformulation of the relative Manin-Mumford problem for semi-abelian schemes, using special subvarieties arising from the mixed Shimura interpretation. The paper presents it as a plausible conjectural statement after correcting the original formulation.