The Galois-orbit lower-bound conjecture for CM points

Let $A_{g,1}$ be the moduli space of principally polarized complex Abelian varieties of dimension $g$. Let $x$ be a CM point in $A_{g,1}$ corresponding to a principally polarized Abelian variety $A_x$, and let $R_x$ be the center of the endomorphism ring $\operatorname{End}(A_x)$. Lower-bound conjecture for CM points. There exists a positive constant $\delta_g$ such that, for all $\epsilon>0$, as $x$ varies over all CM points in $A_{g,1}$,

$$|\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\cdot x|\gg_g \operatorname{Disc}(R_x)^{\delta_g}.$$

The conjecture asks for uniform polynomial lower bounds on Galois orbits in terms of the discriminant of the center of the endomorphism ring. It is recalled here as a central motivation for the higher-rank application, and the source gives no resolution.