Linear Galois-orbit conjecture for abelian varieties and powers of Y(1)

Let $K$ be a field finitely generated over $\mathbb{Q}$. Suppose $X$ is an abelian variety defined over $K$ or $X=Y(1)^n$, and let $V\subseteq X$ be a subvariety defined over $K$. For an algebraic point $P\in V(\overline K)$ that is an optimal singleton of $V$, let $\langle P\rangle$ be the smallest special subvariety containing $P$ and let $\Delta(\langle P\rangle)$ denote its complexity. The property $LGO$ requires bounds of the form

$$\Delta(\langle P\rangle)\leq (2[K(P):K])^{\kappa}$$

for suitable $\kappa>0$, uniformly in the relevant dimensions. Linear Galois-orbit conjecture. If $X$ is an abelian variety defined over $K$ or $X=Y(1)^n$, then $X$ satisfies $LGO$. The conjecture is a quantitative Galois-orbit assertion used to control the complexity of optimal special points; no resolution is stated in the supplied text.