If we consider the set of all unramified Satake parameters $S$ of all automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ as $n$ varies, do we know absolute (that is, independent of $n$) lower and upper bounds for the norm of elements of $S$?

Motivation: denoting by $\mathcal{M}_{a,b}$ the set of such automorphic representations whose set $S_{a,b}$ of Satake parameters at unramified primes $p$ fulfill $a\leq\vert\alpha_{p,j}\vert\leq b$, a proof that all the relevant Satake parameters are bounded above by a quantity independent of $n$ (and $p$ as well) would imply that if $\mathcal{M}_{a,b}$ is closed under the Rankin-Selberg convolution, then $a=b=1$.