Uniform discrete-spectrum bound conjecture

Let $\mathbf G$ be a connected reductive linear algebraic group over $\mathbb Q$, let $\eta:\mathbf G\to\operatorname{GL}(n)$ be a faithful rational representation, and let $\mathcal T$ be a family of bounded depth in $\mathbf G(\mathbb Q)$. For $\Gamma\in\mathcal T$, let $N_\Gamma^{\mathrm{disc}}(\lambda;\sigma)$ count the full discrete spectrum and let $\Gamma_0=\Gamma_n(1)\cap\mathbf G(\mathbb Q)$. Uniform discrete-spectrum bound conjecture. The estimate

$$N_\Gamma^{\mathrm{disc}}(\lambda;\sigma)\leq C[\Gamma_0:\Gamma](1+\lambda)^{d/2}$$

holds uniformly for $\Gamma\in\mathcal T$ and $\lambda\geq0$. The preceding theorem establishes this estimate for the cuspidal spectrum; the source gives no evidence that the discrete-spectrum extension has been proved.