Variance conjecture for closed geodesics in annuli

Let $D$ be a squarefree fundamental discriminant. For $0\leq r<R$, let $A_{r,R}(w)$ be the hyperbolic annulus centered at $w$ with inner radius $r$ and outer radius $R$, and let $\operatorname{Var}(r,R;\Lambda_D)$ denote the variance of the total length of the closed geodesics in $\Lambda_D$ intersecting this annulus. Let $\mathbf{G}$ be the special function arising in the probabilistic model. Variance conjecture. For every fixed $\delta>0$, if

$$0\leq r<R\leq D^{-\delta},$$

then, as $D\to\infty$ through squarefree fundamental discriminants,

$$\operatorname{Var}(r,R;\Lambda_D)\sim \frac{64\sqrt{D}L(1,\chi_D)R^3}{\pi}\mathbf{G}\left(\frac rR\right).$$

The conjecture predicts the variance of closed-geodesic lengths in small annuli from the independent-random-geodesic model; the ball case $r=0$ is the preceding special case. The paper derives the corresponding asymptotic for the probabilistic model, while the assertion for the arithmetic family of closed geodesics remains open.