Large-dimensional hyperbolic spectral exponential sum conjecture

Let $n\geq 2$, let $\Gamma$ be an essentially cuspidal cofinite lattice in $\operatorname{SO}^{+}(1,n)$, and let

$$S(T,X)=\sum_{|t_j|\leq T}X^{it_j}.$$

Hyperbolic spectral exponential sum conjecture. As $X,T\to\infty$, for every $\epsilon>0$,

$$S(T,X)=O\left(X^{\epsilon}T^{n-1+\epsilon}\right).$$

The source presents this as an explicitly formulated conjecture that may be well known to experts, motivated by the modular-surface and Picard-manifold cases. It gives no evidence of a resolution.