Pink's finite-rank intersection conjecture

Let $G$ be a complex semi-abelian variety, let $X$ be a closed irreducible proper subvariety of $G$ of dimension $d$, and let $\Gamma$ be a finite-rank subgroup of $G$. Write $G^{[d+1]}$ for the union of the algebraic subgroups of $G$ of codimension at least $d+1$.

Pink's finite-rank conjecture. If $X$ is not contained in a translate of a proper algebraic subgroup of $G$, then

$$X\cap\bigl(\Gamma\cdot G^{[d+1]}\bigr)$$

is not Zariski dense in $X$.

This formulation is equivalent to Pink's preceding conjecture by adjoining generators of the finite-rank subgroup. It extends the conjectural unlikely-intersection statement from algebraic subgroups to their finite-rank translates and is open in the stated generality.