Reznikov's Lindelöf-type conjecture for geodesic period integrals

Let $(M,g)$ be a compact hyperbolic surface, let $cgamma_{per}$ be a periodic geodesic, and let $e_\lambda$ be an eigenfunction of $\sqrt{-\Delta_g}$ with eigenvalue $\lambda$, parametrized along $cgamma_{per}$ by arc length $s$. Given $\varepsilon>0$, there should exist a constant $C_\varepsilon$ depending on $\varepsilon$, $M$, and the length of $cgamma_{per}$ such that

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

Reznikov's Lindelöf-type conjecture. The geodesic period integrals on compact hyperbolic surfaces satisfy the estimate above. This would improve the known uniform bound and is presented as an analogue of the Lindelöf conjecture for certain $L$-functions; its status is unresolved in the source.