Uniform boundedness for Brauer groups in families

Let $k$ be a finitely generated field of characteristic zero, let $\ell$ be a prime, let $S$ be a quasi-projective variety over $k$, and let $\pi:X\rightarrow S$ be a smooth projective morphism. For every integer $C\geq 1$, let $S^{\leq C}(k)$ be the set of points $s\in S(k)$ for which the image of $\operatorname{Gal}(\overline{k}/k)$ acting on $H^2(X_{\overline{s}},\mathbb{Q}_\ell)$ is open of index at most $C$ in the image of $\pi_1(S)$, and let $S^{\leq C,\mathrm{ex}}(k)$ be its complement in $S(k)$. Galois-theoretic conjecture. There exists an integer $C\geq 1$ such that $S^{\leq C,\mathrm{ex}}(k)$ is not Zariski-dense in $S$. This conjecture is presented as a consequence that would imply the stated uniform boundedness question for Brauer groups, assuming that the fibers over $k$-points satisfy the $\ell$-adic Tate conjecture for divisors. It is described as closely related, for $d=1$, to the Bombieri–Lang conjecture.