Zilber's conjecture on intersections with tori

Let $\mathcal{Y}$ be an irreducible subvariety of ${\bf G}_m^n$ defined over ${\bf C}$. For algebraic subgroups $H\subset {\bf G}_m^n$ satisfying

$$\dim \mathcal{Y}+\dim H\le n-1,$$

consider the union of the intersections $\mathcal{Y}\cap H$. Zilber's conjecture on intersections with tori. If

$$\bigcup_{\substack{H\subset {\bf G}_m^n\\ \dim \mathcal{Y}+\dim H\le n-1}}\mathcal{Y}\cap H$$

is Zariski dense in $\mathcal{Y}$, then $\mathcal{Y}$ is contained in a proper algebraic subgroup of ${\bf G}_m^n$. Zilber's conjecture is stated more generally for semi-abelian varieties, and the conjecture above is open, although many partial results are known.