Conjecture for moments of the partial Hadamard product

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

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

For the partial Hadamard product, set $Z_X(\chi_D)=Z_X(\tfrac12,\chi_D)$. Let $0<c<2$, suppose that $X\leq(2-c)\log g/\log q$, and let $X,g\to\infty$. Partial Hadamard-product moment conjecture. For any $k\geq0$,

$$\left\langle Z_X(\chi_D)^k\right\rangle_{\mathcal{H}_{2g+1}}\sim\frac{G(k+1)\sqrt{\Gamma(k+1)}}{\sqrt{G(2k+1)\Gamma(2k+1)}}\left(\frac{2g}{e^\gamma X}\right)^{k(k+1)/2}.$$

This prediction is supported by a random-matrix model for the zeros and is verified in the paper for $k=1,2,3$; the general case remains open.