Dickson–Hardy–Littlewood prime tuples conjecture for two linear forms

Let $W\geq 1$ be fixed, and let $b_1,b_2$ be fixed distinct integers coprime to $W$. For each prime $p$, let $\nu_p(W\cdot+b_1,W\cdot+b_2)$ be the number of distinct residue classes modulo $p$ represented by $b_1,b_2$ when $p\nmid W$, and let it be zero when $p\mid W$. Dickson–Hardy–Littlewood prime tuples conjecture. As $X\to\infty$, the number of integers $n\leq X$ for which $Wn+b_1$ and $Wn+b_2$ are both prime satisfies

$$\sim \frac{X}{W\log^2 X}\prod_p\left(1-\frac{1}{p}\right)^{-2}\left(1-\frac{\nu_p(W\cdot+b_1,W\cdot+b_2)}{p}\right).$$

This is the two-linear-form case of the prime tuples conjecture and is used here as a conditional input for the asymptotic count of prime pairs with prescribed local conditions. Its resolution is not known in general.