Large Galois orbits conjecture for hybrid orbits

Let $S$ be a Shimura variety, let $F$ be a field of finite type over the reflex field of $S$, and let $s=[x,1]\in S(F)$. Let $\Sigma^g(x)$ denote the geometric hybrid orbit of $x$, and let $H_f$ be the natural height function. Large Galois orbits conjecture. As $x'$ varies in $\Sigma^g(x)$,

$$[F([x',1]):F]\approx H_f(x').$$

The conjecture supplies the lower bounds on Galois-orbit sizes needed for the Pila–Zannier strategy. The paper proves it when $S$ is of abelian type, while the general case remains open.