Conditional first-moment asymptotic conjecture for Rademacher coefficients

Let $(X(k))_{k \geqslant 1}$ be a sequence of independent random variables uniform on $\{\pm 1\}$, and define $(A(N))_{N \geqslant 0}$ by the paper's formal power series. For any finite subset $\mathcal{K} \subseteq \mathbb{N}$ of positive integers and any function $\varepsilon:\mathcal{K}\to\{\pm 1\}$, consider the conditional expectation obtained by fixing $X(k)=\varepsilon(k)$ for every $k\in\mathcal{K}$. Conditional Rademacher first-moment conjecture. There exists an absolute positive constant $C(\varepsilon)$ such that

$$\mathbb{E}\left[|A(N)|: X(k)=\varepsilon(k) \text{ for all } k\in\mathcal{K}\right] \sim \frac{C(\varepsilon)}{(\log N)^{1/4}} \qquad \text{as } N\to\infty.$$

This is a stronger conditional version of the Rademacher case, motivated by the possibility of an Euler-product description of the asymptotic constant. Its status remains conjectural.