Strong Hardy–Littlewood prime 2-tuple conjecture

Let $A>0$ be an integer. Suppose that $(M_1x+a_1)(M_2x+a_2)$ has no fixed prime divisors as $x$ ranges over the integers, with $M_1,M_2>0$ and $0\leq a_i<M_i$ for $i=1,2$. Strong Hardy–Littlewood prime 2-tuple conjecture. There exists an integer $x<\log^A(M_1M_2+1)$ such that both $M_1x+a_1$ and $M_2x+a_2$ are prime. The paper assumes this strong quantitative form of the Hardy–Littlewood prime $k$-tuple conjecture to establish an NP-completeness result for testing whether a set of integers contains a totient; its resolution is not supplied here.