Average eigenvalue bound for Hecke operators on Hermitian modular forms

Let $K$ be the field and level data used in the paper, let $M_0(K)$ be the cuspidal space, and let $HE(K)$ be its Hecke-eigenform basis. Write

$$d_{n,p}:=\dim M_0(K)=|HE(K)|=H_n(p,1)-1.$$

For a prime $\ell\ne p$, let $T(\ell)$ be the Hecke operator and $\lambda_F(T(\ell))$ the eigenvalue of $T(\ell)$ on $F\in HE(K)$. Average eigenvalue conjecture. For each $\ell\ne p$,

$$\limsup_{p\to\infty}\frac{1}{d_{n,p}}\sum_{F\in HE(K)}|\lambda_F(T(\ell))|\le 2^n\ell^{\frac{n(n+1)}{4}}.$$

The bound expresses the expectation that exceptional CAP forms are negligible in the large-$p$ limit, so that the average Hecke eigenvalue is controlled by the non-CAP spectrum. No resolution is supplied in the source.