The Zilber--Pink conjecture for Shimura varieties

Let $S$ be a Shimura variety. A special subvariety of $S$ is an irreducible component of a Shimura subvariety of $S$. An irreducible subvariety $V\subset S$ is Hodge generic if it is not contained in any special subvariety other than a component of $S$ itself.

Zilber--Pink conjecture. If $V$ is an irreducible Hodge generic subvariety of $S$, then the intersection of $V$ with the special subvarieties of $S$ having codimension greater than $\dim V$ is not Zariski dense in $V$.

This is a general unlikely-intersections conjecture for Shimura varieties. The paper proves particular cases for curves in the moduli space of principally polarised abelian varieties, while the general statement remains open.