Fractal Weyl upper bound for resonances on geometrically finite hyperbolic manifolds

Let $\Gamma$ be a geometrically finite discrete group of isometries of $\mathbb{H}^{n+1}$ such that $X=\Gamma\backslash\mathbb{H}^{n+1}$ is a smooth noncompact manifold. Let $R(X)$ denote the set of eigenvalues and resonances of $X$, counted with multiplicity. Let $K\subset T^*X$ be the set of maximally extended, unit-speed geodesics that are precompact, and let $m$ be the Hausdorff dimension of $K$. Fractal Weyl upper bound. For any $C_0>0$, there is $C_1>0$ such that

$$\#\{\sigma\in R(X):|\sigma-r|\le C_0\}\le C_1r^{(m-1)/2}.$$

This conjecture proposes that the resonance counting function in fixed-radius windows is controlled by the Hausdorff dimension of the trapped set, in analogy with fractal Weyl laws. The supplied context presents it as a proposed next step; no resolution is given here.