The resonance concentration conjecture near the pressure line

Let $N(\Omega):=N_{P(h)}(\Omega)$ be the resonance counting function, and let $\mathcal P_E(s)$ be the topological pressure. Resonance concentration conjecture. For every $\gamma>\mathcal P_E(1)/2$, there exists $\epsilon_0>0$ such that for every $\epsilon<\epsilon_0$,

$$\lim_{h\to0}\frac{N\left([E-h,E+h]-\frac{i h}{2}[\mathcal P_E(1)-\epsilon,\mathcal P_E(1)+\epsilon]\right)}{N([E-h,E+h]-ih[0,\gamma])}=1.$$

The claim asserts that asymptotically almost all resonances in the indicated window concentrate near the pressure line; the source presents it as open.