Zilber's finite-family conjecture for atypical intersections

Let $G$ be a complex semi-abelian variety, and let $X$ be a closed irreducible subvariety of $G$. For a closed equidimensional subvariety $H$ of $G$, call an irreducible component $Y$ of $X\cap H$ atypical if

$$\dim(Y)>\dim(X)+\dim(H)-\dim(G).$$

Zilber's conjecture. There is a finite family $\Phi$ of proper algebraic subgroups of $G$ such that, for every algebraic subgroup $H$ of $G$, every atypical component of $X\cap H$ is contained in one of the members of $\Phi$.

The conjecture predicts uniform control of all atypical intersections by finitely many proper algebraic subgroups. It is motivated by Zilber's work on the analogue of Schanuel's conjecture in differential fields and is part of the theory of unlikely intersections.