Polynomial growth conjecture for automorphic coefficients

Let $(\pi,\pi')$ be fixed automorphic representations, let $\pi_{\lambda_i}$ denote the automorphic representation associated with the spectral parameter $\lambda_i$, and let $b_i$ and $a_{\pi,\pi',\pi_{\lambda_i}}$ be the coefficients defined in the paper, with the relation between them as in the preceding discussion. Polynomial growth conjecture. For any $\varepsilon>0$ there exists a constant $C_\varepsilon>0$ such that

$$b_i\leq C_\varepsilon|\lambda_i|^{-2+\varepsilon},$$

as $|\lambda_i|\to\infty$. Equivalently, for fixed $\pi$ and $\pi'$, the corresponding coefficients satisfy a subpolynomial bound in the spectral parameter. The conjecture would sharpen the proved mean-value estimate and give the expected individual, or subconvexity-type, bound for these automorphic coefficients.