Asymptotics of the first two covariance sign changes

Let $p_1(n)<p_2(n)<p_3(n)$ be the three critical probabilities from the preceding conjecture for the relative covariance in the randomly oriented graph $G(n,p)$. Asymptotic sign-change conjecture. Asymptotically,

$$p_1(n)\sim \frac{c_1}{n},\qquad p_2(n)\sim \frac{c_2}{n}$$

for some constants $c_1$ and $c_2$.