Geometrically finite lattice-point counting conjecture for congruence covers

Let $G=\operatorname{SO}(n,1)^\circ$, let $\rho=\tfrac{n-1}{2}$, and let $\Gamma<G(\mathbb{Z})$ be geometrically finite and Zariski dense, with critical exponent $\delta$. Fix $o\in\mathbb H^n$, let $d$ be hyperbolic distance, and define

$$N(T,q):=\#\{\gamma\in\Gamma(q):d(\gamma(o),o)\le T\}.$$

Geometrically finite lattice-point conjecture. For $T\gg1$,

$$N(T,q)\ll_\epsilon\frac{e^{(\delta+\epsilon)T}}{[\Gamma:\Gamma(q)]}+e^{\rho T}$$

for any $\epsilon>0$. This is posed together with the eigenvalue-counting conjecture and the paper proves that it implies that conjecture; the estimate is not proved in the stated generality.