A conjecture of Pink says that in a mixed Shimura variety, every Hodge-generic point is Galois generic (conjecture 6.8 of "A Combination of the Conjectures of Mordell-Lang and Andr\'e-Oort). One can construct mixed Shimura varieties so that they are families of semi-abelian varieties, so Pink's conjecture should have consequences on semi-abelian varieties. Here is where I am confused.
Set up: Let $(P,X^{+})$ be a connected mixed Shimura data. Given a congruence subgroup $\Gamma$ of $P(\mathbb{Q})^{+}$, let $S:=\Gamma\backslash X^{+}$. Let $s$ be any $\mathbb{C}$-valued point on $S$, let $K$ be a finitely generated field extension of $E^{\mathrm{ab}}$ (the maximal abelian extension of the reflex field), such that $s$ is defined over $K$. If $\Gamma'$ is a normal congruence subgroup of $\Gamma$, then we have a continous homomorphism $\mathrm{Gal}(\overline{K}/K)\rightarrow \Gamma/\Gamma'$. By taking limit over the $\Gamma'$, we get $\mathrm{Gal}(\overline{K}/K)\rightarrow\overline{\Gamma}$, where $\overline{\Gamma}$ is the closure of $\Gamma$ in $P(\mathbb{A}_{f})$ ($\mathbb{A}_{f}$ being the ring of finite adeles of $\mathbb{Q}$).
Pink's conjecture: If $s$ is Hodge-generic in $S$, then the image of $\mathrm{Gal}(\overline{K}/K)\rightarrow\overline{\Gamma}$ is open in $\overline{\Gamma}$.
My confusion: Suppose that $s$ is Hodge-generic in $S$ and that $S$ is a family of semi-abelian varieties. Suppose that $s$ lies on a semi-abelian variety $V\subset S$. I thought that Pink's conjecture would imply that the image of $\mathrm{Gal}(\overline{K}/K)\rightarrow T$ is open in $T$ (where $T = \varprojlim V[n]$, and $V[n]$ is the subgroup of $n$-torsion points of $V$), but Jaquinot and Ribet have a construction of semi-abelian varieties with "deficient points", which in particular are non-torion points that don't satisfy this open image condition.
So: what is the correct formulation of Pink's conjecture on semi-abelian varieties?
