Bui–Keating moment conjecture for Dirichlet $L$-functions over function fields

Let $R\in\mathbb F_q[T]$ be a modulus, let $\phi^*(R)$ be the number of primitive Dirichlet characters modulo $R$, and let $\sum^*_{\chi\bmod R}$ denote summation over primitive characters. Let $d_k$ be the $k$-fold divisor function on monic polynomials, let $\mathcal P$ be the set of monic irreducible polynomials, and let $G$ be the Barnes $G$-function. Define

$$a(k):=\prod_{P\in\mathcal P}\left(1-\frac1{|P|}\right)^{k^2}\sum_{m=0}^{\infty}\frac{d_k(P^m)^2}{|P|^m},$$

and

$$f(k):=\frac{G^2(k+1)}{G(2k+1)}=\prod_{i=0}^{k-1}\frac{i!}{(i+k)!}.$$

Bui–Keating moment conjecture. For all non-negative integers $k$,

$$\frac{1}{\phi^*(R)}\sum_{\chi\bmod R}^{*}\left|L\left(\frac12,\chi\right)\right|^{2k}\sim f(k)a(k)\prod_{P\mid R}\left(\sum_{m=0}^{\infty}\frac{d_k(P^m)^2}{|P|^m}\right)^{-1}(\operatorname{deg}R)^{k^2}$$

as $\operatorname{deg}R\to\infty$. This is the function-field analogue of the conjectured moment formula for primitive Dirichlet $L$-functions and is designed so that the arithmetic and random-matrix factors arise naturally.