Ramanujan–Petersson conjecture for unramified automorphic representations

Let $\pi=\bigotimes'\pi_v$ be an irreducible cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_\mathbb Q)$. At an unramified finite prime $p$, write the Satake parameters of $\pi_p$ as $\alpha_{1,p},\ldots,\alpha_{n,p}$. Ramanujan–Petersson conjecture. For every $p$ such that $\pi_p$ is unramified, one has

$$|\alpha_{j,p}|=1\qquad\text{for all }j\in\{1,\ldots,n\}.$$

This is the stronger bound that the paper assumes when necessary; the supplied text does not state a resolution status for general $n$.