Large Galois orbits lower-bound conjecture for hybrid orbits

Let $S$ be a Shimura variety, let $F/E$ be a finite type extension of the reflex field of $S$, let $s=[x,1]\in S(F)$, and let $\Sigma^g(s)$ be a geometric hybrid orbit. Let $w(x')$ denote the parameter associated with $x'$. Large Galois orbits lower-bound conjecture. As $x'$ varies in $\Sigma^g(x)$,

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

This is described as a reformulation of the large Galois orbits conjecture and is used in the paper's dichotomy strategy. It is known for Shimura varieties of abelian type through the stated technical theorem, but is open in general.