Conjecture on the number of zeta-function zeros in strips for hyperbolic rational maps

Let $f$ be a hyperbolic rational map, let $Z(s)$ denote its dynamical zeta function, let \\{\mu_j\} be the zeros of $Z$ counted with multiplicity, and let $\delta$ be the dimension of the Julia set. Zero-counting conjecture. There exists $\epsilon_0>0$ such that for all $0<\epsilon<\epsilon_0$, there exists $0<C_\epsilon<\infty$ such that

$$\#\left\{\mu_j: |\operatorname{Im}\mu_j|\le r,\ -\epsilon^{-1}<\operatorname{Re}\mu_j<\delta\right\}\ge C_\epsilon^{-1}r^{1+\delta}.$$

The conjecture predicts that the lower bound for zeros in strips has the same order as the known upper bound, improving the previously established $\Omega(r^{1-\epsilon})$ lower bound. Its status is not resolved in the supplied source context.