Transport of Zariski density in compatible collections of $G$-representations

Let $X$ be a connected normal scheme of finite type over $\mathbf{Z}$, let $G$ be a connected reductive group over $\mathbf{Q}$, and let $\{{\rho }_{\mathrm{\ell }}\text{\hspace{0.17em}⁣}:{\pi }_1(X[1\mathrm{/}\mathrm{\ell }])\to G({\mathbf{Q}}_{\mathrm{\ell }}){\}}_{\mathrm{\ell }}$ be a Frobenius-compatible collection of continuous homomorphisms indexed by the primes. Assume $\mathrm{I}\mathrm{m}\mathrm{g}({\rho }_{\mathrm{\ell }})$ is Zariski-dense in $G_{{\mathbf{Q}}_{\mathrm{\ell }}}$ for all $\mathrm{\ell }$ in a nonempty finite set $\mathcal{R}$. We prove that, under certain hypotheses on $\mathcal{R}$ (depending only on $G$), $\mathrm{I}\mathrm{m}\mathrm{g}({\rho }_{\mathrm{\ell }})$ is Zariski-dense in $G_{{\mathbf{Q}}_{\mathrm{\ell }}}$ for all $\mathrm{\ell }$ in a set of Dirichlet density $1$. As an application, we combine this result with a version of Hilbert’s irreducibility theorem and recent work of Klevdal–Patrikis to obtain new information about the “canonical” local systems attached to Shimura varieties not of Abelian type.