Orthogonal sector variance conjecture for norm-form divisor sums over Gaussian ideals

Let $d_\ell(\mathfrak a)$ be the $\ell$-fold divisor function on ideals of $\mathbb Z[i]$, let $\chi_2$ be the completely multiplicative character specified by $\chi_2(\pi)=1$ when the Gaussian prime $\pi$ has the relevant norm-form representation and $\chi_2(\pi)=-1$ otherwise, and define

$$\mathcal N^O_{d_\ell,K;x}(\theta)=\sum_{\substack{\mathfrak a=(\alpha)\ \text{ideal}\N(\mathfrak a)\leq x\theta_{\mathfrak a}\in I_K(\theta)}}d_\ell(\mathfrak a)\left(\frac{1+\chi_2(\alpha)}{2}\right).$$

The orthogonal sector variance conjecture. If $x\leq K^\ell$, then there is a constant $a^O_\ell\in\mathbb Q$, depending on $\ell$, such that

$$\mathrm{Var}\left(\mathcal{N}^O_{d_\ell,K;x}\right)\sim a^O_\ell\frac{x}{4K^{1/2}}\gamma_{d_\ell,2}^O\left(\frac{\log x}{\log K}\right)(\log K)^{2\ell^2-\ell-2}.$$

This conjecture is the orthogonal counterpart of the preceding symplectic sector variance prediction and is motivated by the corresponding function-field variance formula; its status is unresolved in the supplied text.