The additively twisted Linnik–Selberg conjecture over $\mathbb{Q}(i)$

Let $C,D\geq 1$, let $0\ne n\in\mathbb{Z}[i]$, and let $S(n,n,c)$ denote the Gaussian Kloosterman sum. Write $\mathrm{N}(c)$ for the norm and $|c|$ for the complex absolute value. For $\varepsilon>0$, use the notation $A\ll_{\varepsilon}B$ to mean that $A$ is bounded by a constant depending on $\varepsilon$ times $B$.

Additively twisted Linnik–Selberg conjecture. One should have

$$\sum_{\mathrm{N}(c)\leq C}\frac{S(n,n,c)}{\mathrm{N}(c)}e\left(\frac{D}{|c|}\right)\ll_{\varepsilon}(|n|CD)^{\varepsilon}.$$

This would be a Gaussian analogue of the conjectural estimates over $\mathbb{Q}$ used in the Linnik–Selberg method. The source proposes it for future work, and its status over $\mathbb{Q}(i)$ is open.