Splitting conjecture for Euler–Hadamard moment factorization

Let $\mathcal{H}_{2g+1}$ be the set of monic square-free polynomials of degree $2g+1$ over $\mathbb{F}_q[x]$, and define

$$\left\langle F\right\rangle_{\mathcal{H}_{2g+1}}=\frac{1}{|\mathcal{H}_{2g+1}|}\sum_{D\in\mathcal{H}_{2g+1}}F(D).$$

Let $P_X(\chi_D)=P_X(\tfrac12,\chi_D)$ and $Z_X(\chi_D)=Z_X(\tfrac12,\chi_D)$ denote the partial Euler and Hadamard products. If $0<c<2$, $X\leq(2-c)\log g/\log q$, and $X,g\to\infty$, Splitting conjecture. For any $k\geq0$,

$$\left\langle L(\tfrac12,\chi_D)^k\right\rangle_{\mathcal{H}_{2g+1}}\sim\left\langle P_X(\chi_D)^k\right\rangle_{\mathcal{H}_{2g+1}}\left\langle Z_X(\chi_D)^k\right\rangle_{\mathcal{H}_{2g+1}}.$$

The conjecture asserts asymptotic independence of the Euler and Hadamard factors at the level of moments. The preceding results establish the corresponding factorization for $k=1,2,3$, while the general statement remains open.