Specific case of the prime tuplets conjecture

Fix a constant $b$ and integers $r_1,r_2,\ldots,r_\ell\leqslant b\log x$. For a prime $p$, let $w(p)$ be the number of distinct solutions $q\bmod p$ of

$$\prod_{j=1}^{\ell}(qr_j+1)\equiv 0\pmod p.$$

Specific case of prime tuplets conjecture. The number of integers $q$ with $x\leqslant q<2x$ for which every $qr_j+1$ is prime satisfies

$$\#\{q:x\leqslant q<2x,\ \text{each }qr_j+1\text{ prime}\}=(1+o_{b,\ell}(1))\frac{x}{(\log x)^\ell}\prod_p\frac{1-w(p)/p}{(1-1/p)^\ell},$$

where the product is over all primes. This is presented as a special case of the Bateman–Horn conjecture in a precise quantitative form and is used as an input for the paper's asymptotic arguments; its resolution is not stated in the supplied text.