Irreducibility of polarized automorphic Galois representations in infinitely many dimensions

Let $\pi$ be a polarized, regular algebraic, cuspidal automorphic representation of ${\mathrm{GL}}_n({\mathbb{A}}_F)$ where $F$ is totally real or imaginary CM, and let $({\rho }_{\lambda }{)}_{\lambda }$ be its associated compatible system of Galois representations. Suppose that $7\nmid n$ and, if $4\mid n$, then $n=4p$ for some prime number $p$. We prove that there is a Dirichlet density $1$ set of rational primes $\mathcal{L}$ such that whenever $\lambda \mid \mathrm{\ell }$ for some $\mathrm{\ell }\in \mathcal{L}$, then ${\rho }_{\lambda }$ is irreducible.