Edixhoven's lower-bound conjecture for Galois orbits of CM points

Let $A_{g,1}$ be the coarse moduli space of principally polarized Abelian varieties. A $g$-dimensional Abelian variety $B$ is CM if $\operatorname{End}(B)\otimes_{\mathbb{Z}}\mathbb{Q}$ contains a semisimple commutative algebra $R$ over $\mathbb{Q}$ with $[R:\mathbb{Q}]=2g$. Let $x$ be the point of $A_{g,1}(\overline{\mathbb{Q}})$ corresponding to a $g$-dimensional CM principally polarized Abelian variety $B$, and let $Z(\operatorname{End}(B))$ be the center of the endomorphism ring of $B$. Edixhoven's conjecture. For each fixed integer $g$, there exists a constant $\delta_g>0$ such that

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

This conjecture predicts polynomial lower bounds for Galois orbits of CM points in terms of the discriminant of the center of the endomorphism ring, and motivates the paper's study of Brauer--Siegel bounds for arithmetic tori.