Generalized Ramanujan conjecture for automorphic representations

Let $K/\mathbb{Q}$ be a number field, let $\pi\in\mathcal{A}_d(K)$ be a cuspidal automorphic representation with unitary central character, and let $R_\pi$ be the set of prime ideals $\mathfrak{p}$ for which $\pi_\mathfrak{p}$ is ramified. For each prime ideal $\mathfrak{p}$, let $\alpha_\pi(j,\mathfrak{p})$ be the local roots of the Euler factor of $L(s,\pi,K)$.

Generalized Ramanujan conjecture. For every prime $\mathfrak{p}\notin R_\pi$ and every $1\leq j\leq d$, $|\alpha_\pi(j,\mathfrak{p})|=1$, while for every prime $\mathfrak{p}\in R_\pi$ and every $1\leq j\leq d$, $|\alpha_\pi(j,\mathfrak{p})|\leq1$.

This is the conjectural strengthening of the known bounds toward temperedness of unramified local components. The supplied material does not establish a resolution of the conjecture in this generality.