The generalized Ramanujan conjecture for cuspidal automorphic representations of general linear groups

Let $F$ be a number field, let $N\in\mathbb{N}$, and let $\pi\in\mathcal{AR}_{cusp}(GL_n)$. For every place $v$ of $F$, let $\pi_v$ denote the local component of $\pi$, and call it tempered when $\sigma(\pi_v)=2$. Generalized Ramanujan conjecture. The local component $\pi_v$ is tempered at every place $v$ of $F$. This conjecture predicts optimal local matrix-coefficient decay for cuspidal automorphic representations of general linear groups and is open for any $N\geq2$ and any number field $F$.