The unlikely intersections conjecture for generically ordinary subvarieties of simple Shimura varieties

Let $\mathcal S$ be a simple Shimura variety over some base, and let $V,W\subseteq\mathcal S$ be generically ordinary subvarieties of complementary dimension. Unlikely intersections conjecture. The set of points of $V$ that are isogenous to some point of $W$ is Zariski-dense in $\mathcal S$. Moreover, the subset of $V\times W$ consisting of pairs of isogenous points is dense in $V\times W$. This extends unlikely-intersection results for families of elliptic curves, abelian surfaces, and $K3$ surfaces to arbitrary subvarieties of a simple Shimura variety; the supplied text gives no resolution, so the conjecture remains open.