Conjectural asymptotic formula for the first moment of multiplicative chaos coefficients

Let $(A(N))_{N \geqslant 0}$ be defined by the formal power series associated with a sequence $(X(k))_{k \geqslant 1}$ of independent standard complex Gaussians. First-moment asymptotic conjecture. There exists a constant $C > 0$ such that

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

The existence of a limiting distribution for $A(N)$ is unknown, and this conjecture gives the predicted order and leading constant for its first absolute moment. The estimate is consistent with known upper and lower bounds of order $(\log N)^{-1/4}$.