Prime separation conjecture for Hecke eigenvalues of distinct Maass forms

Fix an integer $N\geq 2$. Let $\ell_1,\ell_2,\ldots,\ell_N$ be integers greater than one, and let $\eta_1,\eta_2,\ldots,\eta_N$ be distinct Maass forms, with $\eta_j$ a Maass form for $\operatorname{SL}(\ell_j,\mathbb Z)$ and $\lambda_j(n)$ its $n$th Hecke eigenvalue. Prime separation conjecture. There exists a prime $p$ such that $\lambda_1(p),\lambda_2(p),\ldots,\lambda_N(p)$ are all distinct and nonzero. This is presented as evidence for the uniqueness conjecture, and no resolution is supplied in the paper.