Bateman–Horn conjecture for pairs of linear forms

Let $f_1(t)=a_1t+b_1$ and $f_2(t)=a_2t+b_2$ be linear forms with integers $a_1,a_2,b_1,b_2$ and $a_1,a_2>0$. Set $f=f_1f_2$, and suppose that $f$ is admissible, meaning that it does not vanish identically modulo any prime. For each prime $p$, let $\omega_f(p)$ denote the number of $x\in\mathbb{F}_p$ such that $f(x)=0$. Bateman–Horn conjecture. If $f_1$ and $f_2$ are admissible, then

$$\#\{x\leq X:f_1(x),f_2(x)\text{ prime}\}\sim\frac{C_fX}{(\log X)^2},$$

where

$$C_f:=\prod_{p\text{ prime}}\frac{p(p-\omega_f(p))}{(p-1)^2}.$$

In particular, the infinite product defining $C_f$ converges. This is the two-linear-form case of the Bateman–Horn conjecture; according to the supplied status evidence, the only proven case of Bateman–Horn is the prime number theorem for arithmetic progressions, corresponding to one linear polynomial. Thus this claim is resolved in the database as a known conjectural statement rather than an open case.