Lemke Oliver and Soundararajan bias for sums of two squares

Recently Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy–Littlewood conjectures containing a large secondary term, which fits the data very well. We will discuss the analogous question for consecutive pairs of sums of squares in arithmetic progressions, a bias also appears in the experimental data and we develop a similar heuristic model based on the Hardy–Littlewood conjecture for sums of squares to explain it.
This is joint work with Chantal David, Jungbae Nam and Jeremy Schlitt.