The strong Artin conjecture

Let $L/k$ be a Galois extension of number fields with ${\mathrm {Gal}}(L/k) \cong G$. Let $\rho$ be a complex representation of $G$ of dimension $m$. Strong Artin conjecture. There exists an automorphic representation $\pi(\rho)$ of $\operatorname{GL}_m(\mathbb{A}_k)$ such that the $L$-function $L(s,\rho)$ and $L(s,\pi)$ agree at all but finitely many places. Moreover, if $\rho$ is irreducible, then $\pi$ is cuspidal. The conjecture would realize Artin $L$-functions as automorphic $L$-functions and is needed in the paper for an effective Chebotarev density result; it is known in the specific cases used for some of the paper's applications, but remains open in general.