Zilber's conjecture on unlikely intersections in algebraic tori

Let ${\mathbb G}_{\rm m}^N=(\overline{\mathbb Q}^{\times})^N$ be an algebraic torus, and let $W$ be a subvariety of ${\mathbb G}_{\rm m}^N$. For an algebraic subgroup, call it proper if it is not all of ${\mathbb G}_{\rm m}^N$. Zilber's conjecture. There exists a finite collection $\Xi$ of proper algebraic subgroups of ${\mathbb G}_{\rm m}^N$ such that, if $G$ is a proper algebraic subgroup of ${\mathbb G}_{\rm m}^N$ and $Y$ is an irreducible component of $G\cap W$ satisfying

$$\dim(Y)>\dim(G)-{\operatorname{codim}}(W),$$

then there exists $H\in\Xi$ such that $Y\subset H$.

This conjecture describes the exceptional, or unlikely, intersections of subvarieties with algebraic subgroups. The paper notes that theorems of Bombieri–Zannier and Laurent establish the cases of algebraic subgroups of dimensions one and zero, respectively; the general assertion is presented as open here.