Subpolynomial restriction norm conjecture for geodesic circles

Let $X$ be the compact hyperbolic surface, let $O$ be the inverse image under the natural projection of a fixed geodesic circle in the associated surface $Y$, and let $U_{\Lambda}$ be the corresponding eigenspace with restriction Hermitian form $H_{\Lambda}$. For every $\varepsilon>0$, let $A_{\Lambda,\varepsilon}$ be a constant such that

$$H_{\Lambda}(f)\leq A_{\Lambda,\varepsilon}\lVert f\rVert_{W^{\varepsilon}(X)}^2$$

for all $f\in U_{\Lambda}$.

Subpolynomial restriction norm conjecture. For any fixed $\varepsilon>0$ and every $\epsilon>0$,

$$A_{\Lambda,\varepsilon}\ll |\Lambda|^{\epsilon}.$$

This is suggested by quantum chaos intuition and would give a subpolynomial bound for the norm of the restriction map from the eigenspace to the fixed lifted geodesic circle. The source does not provide evidence that this conjecture has been proved or disproved.