Hecke Orbits on Globally Drinfeld-Uniformized Shimura Varieties Are $p$-adically Nowhere Dense

Fix a projective unitary Shimura datum satisfying Varshavsky's global Drinfeld uniformization hypotheses at a place $v\mid p$ of its reflex field $E$. Let $\mathcal{S}h$ be the base change of the Shimura variety to $\mathbb C_p$ at a sufficiently small neat finite level. We prove that for every clos