Hardy–Littlewood lower-bound conjecture for admissible prime tuples

Let $\mathcal A=\{a_1,\ldots,a_s\}$ be an admissible set of distinct natural numbers, meaning that the associated linear forms satisfy the usual local non-obstruction condition. Choose $b\in\{-1,1\}$. The Hardy–Littlewood lower-bound conjecture. The number of integers $n\leqslant x$ for which

$$n,\ a_1n+b,\ldots,a_sn+b$$

are all prime is

$$\gg_{\mathcal A}\frac{x}{(\log x)^{s+1}}.$$

This is a lower-bound form of the prime-tuples conjecture and is not known in general.