Reznikov's period integral conjecture for closed geodesics on hyperbolic surfaces

Let $(M,g)$ be a compact hyperbolic surface, let $e_\lambda$ be an eigenfunction of $\sqrt{-\Delta_g}$ with eigenvalue $\lambda$, and let $\gamma$ be a periodic geodesic or a geodesic circle on $M$. Given $\varepsilon>0$, let $C_\varepsilon$ depend on $\varepsilon$, $M$, and the length of $\gamma_{per}$. Reznikov's period integral conjecture. One has

$$\left|\int_{\gamma} e_\lambda\,ds\right|\leq C_\varepsilon\lambda^{-\frac12+\varepsilon}.$$

This is an analogue of the Lindelöf conjecture for certain $L$-functions and would substantially improve the uniform boundedness estimate known for periodic geodesics. The supplied text does not indicate whether the conjecture has been resolved.