Density hypothesis for Hecke eigenvalues

Let $n>1$, let $\mathcal{F}_T$ be the family of spherical cusp forms with spectral parameter at most $T$, and let $d$ be the Weyl-law exponent. Fix a prime $p$.

Density hypothesis, Hecke eigenvalue version. For every $l\geq 0$, $T\geq 1$, and $\delta>0$, one has

$$\sum_{\varphi\in\mathcal{F}_T}\left|\lambda_\varphi(p^l)\right|^2\ll_{p,\delta}(Tp^l)^\delta\left(T^d+p^{l(n-1)}\right).$$

This is a higher-rank form of Sarnak's density hypothesis: large Hecke eigenvalues should occur with smaller density. The paper assumes this hypothesis to obtain the optimal exponent $\kappa=1$ for every $n$; it is presented as conjectural and is not resolved here.