Gaussian Primes Conjecture for weighted coordinate sums

Let $r\geqslant 1$, let $\Lambda_r=\mu*(\log)^r$ be the von Mangoldt function of order $r$, and write $\Lambda=\Lambda_1$. Define

$$G_r(x)=\mathop{\sum\sum}_{4k^2+\ell^2\leqslant x}\Lambda_r(k)\Lambda(\ell)\Lambda(4k^2+\ell^2).$$

Gaussian Primes Conjecture. One has

$$G_r(x)\sim crx(\log\sqrt{x})^{r-1},$$

where

$$c=\prod_{p\equiv1(4)}\left(1-\frac{3}{p}\right)\left(1-\frac{1}{p}\right)^{-3}\prod_{p\equiv3(4)}\left(1-\frac{1}{p^2}\right)^{-1}.$$

This is a heuristic asymptotic prediction for Gaussian primes whose coordinates are restricted to primes and almost-primes. The case $r=1$ is described as especially challenging because it requires breaking the parity barrier of sieve theory.