Polynomial field-of-definition bound for Hecke translates

In the Shimura-datum setup, let $S_{\mathbf H,\gamma}$ be the special subvariety associated with $\gamma\in\Omega\subset\mathbf G(\mathbb Q)_+$ and let $N(\gamma)$ be the complexity defined from a faithful representation of $\mathbf G$. Field-of-definition conjecture. For every $\kappa>0$, there exists a constant $C_\kappa$ such that, for every $\gamma\in\Omega$, $S_{\mathbf H,\gamma}$ is defined over a number field of degree at most $C_\kappa N(\gamma)^\kappa$. This bound controls the arithmetic complexity of Hecke translates and is used alongside Galois-orbit estimates in the unlikely-intersection method; the general assertion remains open.