Galois-orbit lower bound for Hecke translates

Let $(\mathbf G,X)$ be a Shimura datum and let $S$ be an associated Shimura variety component. Let $(\mathbf H,X_{\mathbf H})\subset(\mathbf G,X)$ be a sub-Shimura datum, and for each $\gamma\in\mathbf G(\mathbb Q)_+$ let $S_{\mathbf H,\gamma}=\pi(\gamma X_{\mathbf H}^+)\subset S$. Let $\rho:\mathbf G\to\mathbf{GL}_{n,\mathbb Q}$ be faithful, and define

$$N(\gamma)=\max\bigl(\operatorname{denom}\rho(\gamma),\lvert\det\rho(\gamma)\cdot\operatorname{denom}\rho(\gamma)^n\rvert\bigr).$$

For $\Omega\subset\mathbf G(\mathbb Q)_+$ and $\Sigma\subset\bigcup_{\gamma\in\Omega}S_{\mathbf H,\gamma}$, define $N(s)=\min\{N(\gamma):\gamma\in\Omega,\ s\in S_{\mathbf H,\gamma}\}$. Let $V\subset S$ be an irreducible algebraic curve not contained in any proper special subvariety, and let $L$ be a finitely generated characteristic-zero field over which $V$ is defined. Galois-orbit conjecture. There exist constants $C_1,C_2>0$ such that for every $s\in V\cap\Sigma$,

$$\#\bigl(\operatorname{Aut}(\mathbb C/L)\cdot s\bigr)\ge C_1N(s)^{C_2}.$$

This conjectural lower bound is an arithmetic ingredient in the Pila–Zannier strategy for restricted Zilber–Pink statements; the paper establishes it only in selected settings, so it remains open in the stated generality.