The finite-field Bateman–Horn conjecture for several polynomials

Let $\mathbf f(x)=(f_1(x),\dots,f_k(x))\in(\mathbb{F}_q[t][x])^k$ be a $k$-tuple of pairwise distinct nonconstant irreducible separable polynomials monic in $x$, and let $f=\prod_{i=1}^k f_i$. Assume

$$\left|\\{a\in\mathbb{F}_q[t]/(P):f(a)=0\\}\right|<|P|$$

for every $P\in\mathcal P_q$. Define

$$C_q(\mathbf f):=\prod_P\frac{1-|P|^{-1}\left|\\{\alpha\in\mathbb{F}_q[t]/(P):f(\alpha)\equiv0\pmod P\\}\right|}{(1-|P|^{-1})^k}.$$

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

$$\left|\\{g\in\mathbb{F}_q[t]:|g|=X,\\ g\text{ monic},\\ f_i(g)\in\mathcal P(\mathbb{F}_q[t])\text{ for all }i\leq k\\}\right|\approx\frac{C_q(\mathbf f)}{\prod_{i=1}^k\deg_x(f_i)}\frac{X}{(\log_q X)^k}.$$

This is the proposed multi-polynomial function-field extension of the one-polynomial statement and includes function-field analogues of simultaneous prime-value problems such as twin primes. It is open in the stated generality.