Local-Global Conjecture for heights of Kleinian group orbits

Let $\mathbb H^3$ be the upper half-space model of hyperbolic $3$-space, and let $\Gamma$ be a geometrically finite subgroup of $\operatorname{SL}_2(\mathbb Z[\sqrt{-\mathfrak D}])$, where $\mathfrak D$ is squarefree. For \gamma=\left(\begin{matrix}a&b\c&d\end{matrix}\right), define the height by $H(\gamma)=|c|^2+|d|^2$, and let $\delta_\Gamma$ be the critical exponent of $\Gamma$. Denote by $\mathcal A(\Gamma)$ the admissible set associated with $\Gamma$. Local-Global Conjecture. If $\delta_\Gamma>1$, then $H(\Gamma)=\{H(\gamma):\gamma\in\Gamma\}$ and its admissible set $\mathcal A(\Gamma)$ differ by a finite set. The conjecture proposes that, apart from finitely many exceptions, every locally admissible height is globally represented by the orbit.