Anomalous-level variance conjecture for fields with spectral singularity

Let $f$ satisfy Assumptions and, and suppose there exist $\alpha\in(0,2)$ and $r_0>0$ such that

$$\rho(x)=|x|^{-\alpha}$$

for all $|x|<r_0$. Let $D\subset\mathbb{R}^2$ be an open rectangle centred at the origin, and let $N_\mathrm{ES}(D_R,\ell)$ and $N_\mathrm{LS}(D_R,\ell)$ denote the numbers of excursion-set and level-set components. Anomalous-level variance conjecture. There exists a possibly empty finite set $\mathcal{L}\subset\mathbb{R}$ and $c_\mathrm{var}(\ell)>0$ such that, for all $\ell\notin\mathcal{L}$,

$$\mathrm{Var}(N_\mathrm{ES}(D_R,\ell))\sim c_\mathrm{var}(\ell)\mathrm{Area}(D)^{\frac{2+\alpha}{2}}R^{2+\alpha},$$

whereas for all $\ell\in\mathcal{L}$,

$$\mathrm{Var}(N_\mathrm{ES}(D_R,\ell))\ll R^{2+\alpha},$$

and the same conclusion holds for $N_\mathrm{LS}(D_R,\ell)$ with a different set $\mathcal{L}$. If $f$ is the Random Plane Wave, the same conclusion holds with $2+\alpha$ replaced by $3$. The source presents this as an open conjecture describing generic and anomalous levels for singular spectral measures.