The cohomological-dimension and critical-exponent conjecture

Let $\Gamma$ be a Kleinian group in $\operatorname{Mob}(\mathbb S^n)$ without parabolic elements, let $cd(\Gamma)$ denote its cohomological dimension, and let $\delta(\Gamma)$ denote its critical exponent. Cohomological-dimension and critical-exponent conjecture. One has

$$cd(\Gamma)-1\leq \delta(\Gamma).$$

If equality holds, then $\Gamma$ is an $i$-fuchsian convex-cocompact group, where $i=\delta(\Gamma)$. The source gives partial confirmation of the inequality and proves the equality characterization for geometrically finite groups, but does not resolve the full conjecture.