Generic-level variance conjecture for excursion and level sets

Let $f$ satisfy Assumptions,, and, and let $D\subset\mathbb{R}^2$ be an open rectangle centred at the origin. Write $D_R$ for its dilation by $R$, 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 at level $\ell$. Generic-level variance conjecture. For every $\ell\in\mathbb{R}$ there exists $c_\mathrm{var}(\ell)>0$ such that

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

and the same conclusion holds for $N_\mathrm{LS}(D_R,\ell)$. The conjecture asserts that the proved lower bound has the correct order for these fields and all levels; the source presents it among open questions, so its resolution is not given.