The p-adic nowhere density conjecture for Hecke orbits of positive-dimensional subvarieties

Let $\mathcal{S}h$ be a Shimura variety and let $Z\subseteq\mathcal{S}h$ be a positive-dimensional subvariety. Denote by the Hecke orbit of $Z$ the union of its images under the Hecke operators, viewed as a subset of the $p$-adic analytic space $\mathcal{S}h(\mathbb{C}_p)$. The p-adic nowhere density conjecture. The Hecke orbit of $Z$ is $p$-adically nowhere dense in $\mathcal{S}h(\mathbb{C}_p)$. This is a higher-dimensional analogue of the known nowhere-density results for Hecke orbits of points on PEL-type Shimura varieties with good reduction at $p$. The conjecture concerns the geometry of positive-dimensional subvarieties stable under Hecke operators and remains open in the stated generality.