The large Galois orbits conjecture for endomorphism-generic special subvarieties

Let $g\geq 3$, let $C$ be an irreducible algebraic curve in $\mathcal{A}_g$, and let $S$ be the smallest special subvariety of $\mathcal{A}_g$ containing $C$. Let $\Omega$ be the set of special subvarieties of $\mathcal{A}_g$ of simple PEL type I or II and dimension at most $\dim(S)-2$, and let $\Sigma$ be the set of points in $\mathcal{A}_g(\mathbb{C})$ that are endomorphism generic in some $Z\in\Omega$. Let $L$ be a finitely generated subfield of $\mathbb{C}$ over which $C$ is defined.

Large Galois orbits conjecture. There exist positive constants $C_1$ and $C_2$ such that, for every $s\in C\cap\Sigma$,

$$\#\operatorname{Aut}(\mathbb{C}/L)\cdot s\geq C_1\lvert\operatorname{disc}(\operatorname{End}(A_s))\rvert^{C_2}.$$

This is presented as the natural generalisation of the earlier large Galois orbits conjecture and is used to deduce finiteness in a Zilber--Pink setting. Its general status is not resolved in the supplied text.