Absolute irreducibility conjecture for compatible systems attached to automorphic representations

Let $F$ be the number field and $\bar F$ its algebraic closure, with absolute Galois group $\Gamma_F=\operatorname{Gal}(\bar F/F)$. Let $\pi$ be a regular algebraic, conjugate self-dual, cuspidal automorphic representation of $\operatorname{GL}_n(\mathbb{A}_F)$, and let $E_\pi$ be its coefficient field. For each place $\lambda$ of $E_\pi$ above a rational \prime $\ell$, write $E_{\pi,\lambda}$ for the corresponding completion and let

$$\rho_{\pi,\lambda}:\Gamma_F\longrightarrow\operatorname{GL}_n(E_{\pi,\lambda})$$

be the associated Galois representation. Absolute irreducibility conjecture. There exists a set of rational primes $\mathcal{L}$ of Dirichlet density $1$ such that, for all $\ell\in\mathcal{L}$ and every $\lambda\mid\ell$, the representation $\rho_{\pi,\lambda}$ is absolutely irreducible.

This is a weaker consequence of the global Langlands conjectures, which predict absolute irreducibility for all members of the compatible system. Its proof is widely open in general, although partial progress is known.