Pink's Zilber–Pink conjecture for semiabelian varieties

Let $G/\mathbb{C}$ be a semiabelian variety, and for a positive integer $m$ let $G^{[m]}$ denote the union of the complex points of all subgroups of $G$ of codimension at least $m$. Let $X/\mathbb{C}$ be a subvariety of dimension $d$ of $G$, and suppose that $X$ is not contained in a proper abelian subgroup of $G$.

Pink's conjecture. The intersection

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

is not Zariski dense.

This is a semiabelian-variety case of the broader Zilber–Pink conjecture, which includes Mordell–Lang and André–Oort as special cases. The source supplies no resolution evidence for this formulation, so it remains open.