The large Galois orbits conjecture for endomorphism loci in principally polarised abelian varieties

Let $g\geq 3$. Let $\Sigma\subset\mathcal{A}_g$ be the set of points whose associated abelian variety has endomorphism algebra either a totally real field other than $\mathbb{Q}$ or a non-split totally indefinite quaternion algebra over a totally real field. Let $C\subset\mathcal{A}_g$ be an irreducible Hodge generic algebraic curve defined over a finitely generated field $L\subset\mathbb{C}$.

Large Galois orbits conjecture. There exist positive constants $C_1$ and $C_2$, depending only on $g$, $L$, and $C$, 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 conjecture gives the polynomial lower bound on Galois-orbit sizes needed for the stated Zilber--Pink applications; the paper proves it in certain cases, but its general form remains open.