Bateman–Horn conjecture over function fields

Fix an odd prime $p$ and a power $q$ of $p$, and let $\mathbb{F}_q[u]$ be the polynomial ring over the field with $q$ elements. Let $\pi$ range over monic irreducible polynomials in $\mathbb{F}_q[u]$, let $|f|=q^{\deg(f)}$ for nonzero $f\in\mathbb{F}_q[u]$, and let $F(T)\in\mathbb{F}_q[u][T]$ be an irreducible separable monic polynomial. Define

$$\mathfrak{S}_q(F)=\frac{1}{\deg_T(F)}\prod_{\pi}\frac{1-\frac{1}{|\pi|}\#\{x\in\mathbb{F}_q[u]/(\pi):F(x)\equiv0\ \mathrm{mod}\ \pi\}}{1-\frac{1}{|\pi|}}.$$

Function-field Bateman–Horn conjecture. As $X\to\infty$ through powers of $q$,

$$\#\{g\in\mathbb{F}_q[u]:|g|=X,\ g\ \text{is monic},\ F(g)\ \text{is prime}\}\sim\mathfrak{S}_q(F)\cdot\frac{X}{\log_qX}.$$

The paper resolves the quadratic case when $q$ satisfies the stated sufficiently large lower bound, but the conjecture in general degree remains open.