The prime $k$-tuple conjecture predicts that for $(a_{1}n + b_{1}), \ldots, (a_{k}n + b_{k})$ an "admissible" k-tuple, where the $a_{i}, b_{i}$'s are fixed, then there are $$ \sim c \frac{x}{(\log{x})^{k}} $$ integers $n \leq x$ such that all $a_{i}n + b_{i}$ are simultaneously prime.
I cannot find any mention of the $a_{i},b_{i}$'s varying however. That is, what happens (or is expected to happen) when these grow as a function of $x$ as well, say as some small power of $x$? Is there some conjecture, or preliminary results? Essentially I'm wondering if somehow implied by the conjecture or by other work is the fact that if the conjecture holds as stated above then it holds whenever $a_{i}, b_{i} \leq f(x)$ for f some function.
A plausible formula is given by the Bateman-Horn conjecture.