Cohomological Sarnak–Xue density hypothesis

Let $G$ be as above, let $\Pi^{\mathrm{alg}}(G)$ be the finite-dimensional algebraic representations of $G$, and let $\Pi^{\mathrm{coh}}(G_\infty;E)$ denote the irreducible representations with nonzero $(\mathfrak g,K_\infty)$-cohomology with coefficients in $E$. Write $m(\pi;q)$ for the relevant automorphic multiplicity. Cohomological Sarnak–Xue density hypothesis. For any $E\in\Pi^{\mathrm{alg}}(G)$ and any $\pi\in\Pi^{\mathrm{coh}}(G_\infty;E)$,

$$m(\pi;q)\ll\operatorname{vol}(\Gamma(q)\backslash G_\infty)^{\frac{2}{r(\pi)}}.$$

This is presented as a special case of the Sarnak–Xue density hypothesis; the source gives no resolution.