The prime-to-p Hecke orbit conjecture for K3 moduli strata

Let $\shS^i$ be the height and Artin-invariant strata in the special fiber of the orthogonal Shimura variety parametrizing degree-$2d$ polarized K3 surfaces, and let the prime-to-$p$ Hecke orbit of a point be defined using the $G(\bA_f^p)$-orbit of a lift to the corresponding hyperspecial-level Shimura variety. The prime-to-$p$ Hecke orbit conjecture. For $1\le i\le 10$ or $i=20$, the prime-to-$p$ Hecke orbit of every $s\in\shS^i(\bar{\bF}_p)$ is Zariski dense in $\shS^i$. This predicts density of prime-to-$p$ Hecke orbits in the indicated ordinary-height and superspecial strata; the supplied text gives no evidence that the prediction has been resolved.