The one-log-zone conjecture for resonance-free regions

Let $M$ be a cusp manifold and let $\mathcal{G}(M)$ denote the space of $C^\infty$ metrics on $M$ for which $(M,g)$ is a cusp manifold with negative sectional curvature. The theorem establishes a resonance finiteness region for a bigger open and dense set of metrics $g\in\mathcal{G}(M)$, namely those for which there are constants $\delta>0$ and $C_0>0$ such that

$$\left\{s\in\operatorname{Res}(M,g),\ \Re s<d-\delta,\ \Re s>-C_0\log|\Im s|\right\}$$

is finite. One-log-zone conjecture. The set of metrics with this property is actually all of $\mathcal{G}(M)$. This would extend the theorem from an open and dense set of negatively curved cusp metrics to every such metric.