Ramanujan's conjecture for half-integral weight automorphic forms

Let $F$ be a totally real field, let $S_{\infty}$ be the set of its archimedean places, and let $\tilde{A}_{00}$ denote the relevant cuspidal automorphic space. Let $\tilde{\pi}$ be an irreducible subrepresentation of $\tilde{A}_{00}$, let $\tilde{\varphi}\in V_{\tilde{\pi}}$, and let $D\in F^*$ be a square-free integer. Write $|D|_{S_{\infty}}=\prod_{v\in S_{\infty}}|D|_v$ and let $d_{\tilde{\pi}}(\tilde{\varphi},S_{\infty},\psi^D)$ be the normalized Fourier coefficient defined in the paper.

Ramanujan conjecture. As $|D|\mapsto\infty$, for every $\alpha>0$,

$$|d_{\tilde{\pi}}(\tilde{\varphi},S_{\infty},\psi^D)|\ll_{\tilde{\pi},\tilde{\varphi},\alpha}|D|_{S_{\infty}}^{\alpha-1/2},$$

where the implied constant depends only on $\tilde{\pi}$, $\tilde{\varphi}$, and $\alpha$. The paper states that this conjecture implies the corresponding bound for Fourier coefficients of half-integral weight forms, while its general status is not resolved.