Sarnak's Weyl-law conjecture for congruence subgroups and arbitrary types

Let $G$ be the reductive group considered in the paper, let $\Gamma\subset G(\mathbb{Q})$ be a congruence subgroup, and let $\Pi(K_\infty)$ denote the set of $K_\infty$-types. For $\nu\in\Pi(K_\infty)$, let $N_{\Gamma,\mathrm{cus}}(\lambda;\nu)$ and $N_{\Gamma,\mathrm{res}}(\lambda;\nu)$ denote the counting functions of the cuspidal and residual spectra of type $\nu$. Sarnak's conjecture. For every $\nu\in\Pi(K_\infty)$, $N_{\Gamma,\mathrm{cus}}(\lambda;\nu)$ satisfies Weyl's law and $N_{\Gamma,\mathrm{res}}(\lambda;\nu)$ is of lower-order growth. This is the higher-rank arithmetic formulation of the expected spectral asymptotics: the cuspidal spectrum has the full Weyl-law growth, while the residual spectrum is negligible to leading order. The source presents this as an open conjecture.