Extension of the first-moment asymptotic conjecture to real Gaussian and Rademacher coefficients

Let $(A(N))_{N \geqslant 0}$ be defined by the same formal power series from a sequence $(X(k))_{k \geqslant 1}$. Real-Gaussian and Rademacher extension conjecture. The first-moment asymptotic formula

$$\mathbb{E}[|A(N)|] \sim \frac{C}{(\log N)^{1/4}} \qquad \text{as } N \to \infty$$

should also hold when $(X(k))_{k \geqslant 1}$ is a sequence of independent standard real Gaussians or a sequence of independent random variables uniform on $\{\pm 1\}$, with a possibly different constant $C$ in each case. This extends the complex-Gaussian conjecture to two other coefficient distributions; the paper explains that the known moment estimate is plausible in these settings and investigates them computationally.