Generalized relative Manin–Mumford conjecture

Let $\pi:A\rightarrow S$ be an Abelian scheme over an irreducible complex variety $S$, and let $Y\subset A$ be irreducible and dominate $S$. Let $Y_{\mathrm{tor}}^{\geq r}$ denote the set of irreducible components of $Y_{\mathrm{tor}}$ of dimension at least $r$. Suppose that

$$r+\dim A > \dim Y+\dim S.$$

If $Y_{\mathrm{tor}}^{\geq r}$ is Zariski dense in $Y$, then $Y$ is contained in a proper torsion coset.

Generalized relative Manin–Mumford conjecture. Under these hypotheses, $Y$ is contained in a proper torsion coset.

This is presented as a generalized version of the Relative Manin–Mumford conjecture, with the case $r=0$ identified with that conjecture. The source does not specify a resolution status for the generalized statement.