Asymptotic conjecture for successive sums of two squares in arithmetic progressions

Fix a prime $q\equiv1\,(\mathrm{mod}\,4)$, an integer $r\geq2$, and $\mathbf{a}=(a_1,\dots,a_r)\in\mathbb{N}^r$. Let $E_n$ be the increasing sequence of sums of two squares, and define

$$N(x;q,\mathbf{a})=\#\{E_n\leq x:E_{n+i-1}\equiv a_i\,(\mathrm{mod}\,q)\}.$$

Let $K$, $C_{a_i,a_{i+1}}$, and $C_1$ be the constants defined in the source. Conjecture on successive sums of two squares. The stated asymptotic expansion with coefficients $C_{-1}(\mathbf a)$, $C_0(\mathbf a)$, and $C_1(\mathbf a)$ holds exactly as displayed in the source statement. This heuristic prediction extends the two-term bias model to $r$ successive sums of two squares; it is open.