The complete positive bias conjecture for Gaussian-prime representations

Let

$$D_2(x)=\lvert\{p<x:p=a^2+4b^2,\ \lvert a\rvert\equiv1\bmod4\}\rvert-\lvert\{p<x:p=a^2+4b^2,\ \lvert a\rvert\equiv3\bmod4\}\rvert.$$

Here $p$ ranges over primes, and logarithmic scale is measured by logarithmic density, using the variable $y=\log x$; thus a property holds for almost all $x$ in logarithmic scale when the corresponding set has logarithmic density one. Complete positive bias conjecture. The values of $D_2$ have a complete bias towards positive values: for almost all $x\in[2,\infty)$ in logarithmic scale, more than half of the primes below $x$ have a representation $p=a^2+4b^2$ with $\lvert a\rvert\equiv1\bmod4$. Precisely,

$$D_2(x)\geq0$$

for almost all $x$ in logarithmic scale, and

$$D_2(x)=\Omega_{+}\left(\frac{\sqrt{x}}{\log x}\right).$$

This conjecture predicts a one-sided prime-number race for the two congruence classes of the odd square parameter in representations $p=a^2+4b^2$. It is part of the paper's heuristic model for the distribution of $D_2$; the source supplies no proof or resolution.