Weaken Dickson's conjecture

Question

I'm interesting in Dickson's conjecture. But it is hard to consider. I want some `weaken' Dickson's conjecture. It is the following statement.

Let $a$ and $b$ be constant integers. $S$ is a set of infinite primes. Then, the following statements are equivlant.

1. There are infinitely many integers $n$ such that $n+a$ and $n+b$ have only prime factors in $S$
2. There is an integer $n$ such that $n+a$ and $n+b$ have only prime factors in $S$

Is there a result for this weaker statement?