Poisson variance conjecture for shrinking spherical sets

Let $\mathcal E(n)$ be the projected lattice points on $S^2$, let $N_n$ be their number, and let $\sigma$ denote normalized spherical area. For a spherical cap or annulus $\Omega_n\subset S^2$, let $Z(n;g\Omega_n)$ count the projected lattice points in $g\Omega_n$, where $g\in\operatorname{SO}(3)$ and $dg$ is Haar probability measure. Poisson variance conjecture. If

$$N_n^{-1+\epsilon}\ll\sigma(\Omega_n)\ll N_n^{-\epsilon}$$

as $n\to\infty$, with $n\neq0,4,7\bmod8$, then

$$\int_{\operatorname{SO}(3)}\left|Z(n;g\Omega_n)-N_n\sigma(\Omega_n)\right|^2\,dg\sim N_n\sigma(\Omega_n).$$

For random points, the variance equals the expected number of points, motivating this prediction. Under the Lindelöf hypothesis for standard $GL(2)/\mathbb Q$ $L$-functions, the paper proves an upper bound with an $n^\epsilon$ loss; the asymptotic formula remains open.