Irreducibility of polarized automorphic Galois representations in infinitely many dimensions

Let \( π\) be a polarized, regular algebraic, cuspidal automorphic representation of \( \GL_n(\bb{A}_F) \) where \( F \) is totally real or imaginary CM, and let \( (ρ_λ)_λ\) be its associated compatible system of Galois representations. We prove that if \( 7\nmid n \) and \( 4 \nmid n \) then there is a Dirichlet density \( 1 \) set of rational primes \( \mc{L} \) such that whenever \( λ\mid \ell \) for some \( \ell\in \mc{L} \), then \( ρ_λ\) is irreducible.