André–Oort equidistribution conjecture for products of modular curves

Let $X$ be a finite product of complex modular curves. Let $\{x_i\}_i$ be a sequence of special points on $X$, meaning that every coordinate of $x_i$ is a CM point, and let $\mu_i$ be the normalized counting measure on the finite Galois orbit of $x_i$. If $\{x_i\}_i$ has finite intersection with every proper special subvariety of $X$, then $\{\mu_i\}_i$ converges weak-$*$ to the uniform probability measure on $X$.

Equidistribution conjecture. Under the stated finiteness condition, the Galois-orbit measures $\mu_i$ converge weak-$*$ to the uniform probability measure on $X$.

This conjecture is a special case of the equidistribution conjecture for Galois orbits of special points and implies the André–Oort conjecture for products of modular curves. Its status is not resolved in the source.