CFKRS moment conjecture for the random Euler-product model of function-field L-functions

Let $r$ and $\tilde{r}$ be nonnegative integers, let $q>2$ be a prime power, let $s_1,\dots,s_{r+\tilde{r}}$ be complex numbers with real part $\frac{1}{2}$, and let $A$ be a real number. Let $L$ range over $\mathbb C[[q^{-s}]]^+$, let $\mu_{\textrm{ch}}$ be the measure on this space, and let $\operatorname{MT}_{N}^{r,\tilde{r}}(s_1,\dots,s_{r+\tilde{r}})$ denote the CFKRS main term. CFKRS moment conjecture. The moment satisfies

$$\int_{\mathbb C[[q^{-s}]]^+} \Bigl( \prod_{j=1}^{r} L(s_j) \prod_{j=r+1}^{r+\tilde{r}} \overline{L(s_j)} \Bigr)\,\mu_{\textrm{ch}} = \operatorname{MT}_{N}^{r,\tilde{r}}(s_1,\dots,s_{r+\tilde{r}}) + O_{q,r,\tilde{r},A}(N^{-A}).$$

If true, this would make $\mu_{\textrm{ch}}$ reproduce every coefficient of the polynomial CFKRS main term, strengthening the preceding power-saving estimate and supporting the random-matrix model for the family of function-field $L$-functions. The conjecture is presented as an expected strengthening; no resolution is given in the supplied text.