The Large Galois Orbits conjecture for endomorphism-generic PEL 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 III or IV and dimension at most $\dim(S)-2$. Let $\Sigma$ be the set of points in $\mathcal{A}_g(\mathbb{C})$ that are endomorphism generic in some member of $\Omega$. Let $L\subset\mathbb{C}$ be a finitely generated subfield over which $C$ is defined, and write $A_s$ for the abelian variety associated with $s$.

Large Galois Orbits conjecture. There exist positive constants $C_{\mathrm{mult}}$ and $C_{\mathrm{exp}}$ such that, for all $s\in C\cap\Sigma$,

$$\#\operatorname{Aut}(\mathbb{C}/L)\cdot s\geq C_{\mathrm{mult}}\,\bigl|\operatorname{disc}(\operatorname{End}(A_s))\bigr|^{C_{\mathrm{exp}}}.$$

This is presented as the natural generalisation of the earlier large Galois-orbit conjecture. It is used to obtain finiteness for the intersection $C\cap\Sigma$, but no resolution is supplied in the text.