Mazur's algebraic equidistribution conjecture for Hecke orbits

Let $S_K(G,X)$ be the Shimura variety under consideration, and let $Y\subset S_K(G,X)$ be a positive-dimensional subvariety intersecting the ordinary locus. The Hecke orbit of $Y$ is said to equidistribute when, for every sequence of prime-to-$p$ Hecke correspondences $\tau_i$ with $\deg\tau_i\to\infty$, the minimal degree of a hypersurface containing $\tau_i(Y)$ tends to infinity. Mazur's algebraic equidistribution conjecture. If $Y\subset S_K(G,X)$ is a positive-dimensional subvariety that intersects the ordinary locus, then the Hecke orbit of $Y$ equidistributes. This is an algebraic positive-characteristic analogue of the equidistribution of Hecke orbits used in the characteristic-zero André–Pink–Zannier argument; the source presents it as a reasonable conjecture because no general positive-characteristic analogue of that analytic equidistribution is known.