Pink's intersection conjecture for semi-abelian varieties

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

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

$$X\cap G^{[d+1]}$$

is not Zariski dense in $X$.

This is a natural generalization of the theorem of Maurin and extends the Manin–Mumford setting from torsion points to intersections with algebraic subgroups of sufficiently large codimension. The statement is presented as a conjecture of Pink and remains open in this generality.