The fractal Weyl law for resonance lower bounds

Let $N(\Omega):=N_{P(h)}(\Omega)$ be the resonance counting function, and let $\delta_E$ be the trapped-set dimension parameter defined in the source. Fractal Weyl law conjecture. If $\dim M=2$, then

$$\lim_{h\to0}h^{\delta_E}N([E-h,E+h]-ih[0,\gamma])=v_E(\gamma),$$

where $v_E(\gamma)>0$ for sufficiently large $\gamma$. If $\dim M>2$, then

$$\limsup_{h\to0}h^{\delta_E}N([E-h,E+h]-ih[0,\gamma])>0$$

for sufficiently large $\gamma$. This conjecture seeks a lower bound matching the fractal Weyl upper bound; the source says that only special toy models had been proved.