Pink's original relative Manin–Mumford conjecture for semi-abelian schemes

Let $S$ be a variety over $\mathbb{C}$, and let $\mathcal{G}\to S$ be an algebraic family of semi-abelian varieties. Let $Z\subset\mathcal{G}$ be an irreducible closed subvariety of dimension $d$ that is not contained in any proper closed subgroup scheme, possibly over a smaller base, of $\mathcal{G}\to S$. Define $\mathcal{G}^{[>d]}$ fiberwise as the union of subgroups of codimension greater than $d$.

Pink's original relative Manin–Mumford conjecture. The intersection $Z\cap\mathcal{G}^{[>d]}$ is not Zariski dense in $Z$.

The source explicitly reports a counterexample when the family has nonzero toric rank, arising from a Ribet section. Thus the original formulation is refuted.