The pressure threshold conjecture for resonance density

Let $N(\Omega):=N_{P(h)}(\Omega)$ be the resonance counting function, and let $\mathcal P_E(s)$ denote the topological pressure from the source. Pressure threshold conjecture. For general operators satisfying the source's section assumptions,

$$\gamma<\frac12|\mathcal P_E(1)|\quad\Longrightarrow\quad v_E(\gamma)=0.$$

For convex co-compact quotients this conclusion follows from the cited theorem, but the source formulates it as a conjecture for general operators.