Reznikov's Fourier coefficient conjecture for restrictions to geodesics

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 closed geodesic or a geodesic circle, parametrized by arc length as $\gamma_{per}(s)$. Given $\varepsilon>0$ and $0<c<1$, let $C_\varepsilon$ depend on $\varepsilon$, $M$, and the length of $\gamma_{per}$. Reznikov's Fourier coefficient conjecture. Whenever $0\leq\nu/\lambda\leq c<1$, one has

$$\left|\int_{\gamma_{per}}e_\lambda(\gamma_{per}(s))e^{-i\nu s}\,ds\right|\leq C_\varepsilon\lambda^{-\frac12+\varepsilon}.$$

This conjecture predicts power savings over the logarithmic Fourier-coefficient estimate stated immediately before it. The supplied text gives no evidence of resolution.