Hardy–Littlewood conjecture for sums of two squares in arithmetic progressions

For $k\geq 1$, let $\mathcal{H}=\{h_1,\dots,h_k\}\subseteq\mathbb{Z}$, let $q\equiv1\,(\mathrm{mod}\,4)$ be prime, let $a\in\mathbb{Z}$, and let $R_k(\mathcal{H};x,q,a)$ be the normalized count of integers $n\leq x$ with $n\equiv a\,(\mathrm{mod}\,q)$ for which every $n+h_i$ is a sum of two squares. Let $\mathfrak{S}(\mathcal{H})$ be the singular series. Hardy–Littlewood conjecture in arithmetic progressions. If $\mathfrak{S}(\mathcal{H})>0$, then

$$R_k(\mathcal{H};x,q,a)\sim\frac{\mathfrak{S}(\mathcal{H})}{q}\left(\frac{K}{\sqrt{\log x}}\right)^k.$$

This is a proposed arithmetic-progression analogue of the Hardy–Littlewood conjecture and remains open.