Jakobson–Naud's half-plane conjecture for zeros of the Selberg zeta function

Let $Z_X(s)$ be the Selberg zeta function of the symmetric infinite-area hyperbolic surface under consideration, and let $b4$ denote the exponent of convergence of its limit set.

Jakobson–Naud's conjecture. There are only finitely many zeros in the half-plane

$$\Re(s)>\frac{\delta}{2}$$

and this is the largest half-plane with this property: for every $\varepsilon>0$, there are infinitely many zeros in the half-plane

$$\Re(s)>\frac{\delta}{2}-\varepsilon.$$

The conjecture identifies the boundary $\Re(s)=\delta/2$ as the sharp finiteness threshold for zeros of the Selberg zeta function. It is attributed to Jakobson and Naud and is presented here as an open conjecture.