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It is generally believed that the Second Hardy-Littlewood Conjecture is false. But it has not been proved (or disproved) yet. My question is,

What would be the consequences if Second Hardy-Littlewood Conjecture turns out to be true?

Obviously one consequence is that the First Hardy-Littlewood Conjecture (or the $k$-tuple Conjecture) would be false and hence all the results that would follow iff the $k$-tuple Conjecture would hold will be false. To be honest I don't know any such conjectures which holds iff the $k$-tuple Conjecture holds. So it will be great if someone provides some references regarding this issue.

Apart from this results or examples, if there are some conjectures which would follow iff the Second Hardy-Littlewood Conjecture is true, a reference of them would be nice.

Added:- Due to the comment below, I will briefly state the two conjectures. But I retain the links for a reference to some authentic source.

$k$-tuple Conjecture

If $b_1,b_2,\ldots,b_k$ is an admissible $k$-tuple of integers then there exists infinitely many integral values of $x$ such that $x+b_1,x+b_2,\ldots, x+b_k$ are all prime.

Here by the admissible $k$-tuple $b_1,b_2,\ldots,b_k$ we mean that for every prime $p$ there exists an integer $x$ such that all the integers $x+b_1,x+b_2,\ldots, x+b_k$ are incongruent modulo $p$.

Second Hardy-Littlewood Conjeture

$$\pi(x)+\pi(y)\ge \pi(x+y)\qquad \forall x,y\geq 2$$

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    $\begingroup$ It would help if you could quote the conjecture in the question to make it self-contained. $\endgroup$
    – Mark Bennet
    Commented Dec 31, 2014 at 14:48
  • $\begingroup$ In the answer to math.stackexchange.com/questions/1072194/… the source of the second conjecture in Hardy's paper is given. $\endgroup$
    – daniel
    Commented Dec 31, 2014 at 15:41
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    $\begingroup$ You might be interested in this answer of Terry Tao to a related question, which says "I don't recommend putting too much time into taking random conjectures in number theory and trying to figure out what they imply": mathoverflow.net/a/190892/25028 $\endgroup$
    – Sam Hopkins
    Commented Jan 5, 2015 at 18:05
  • $\begingroup$ @SamHopkins: Nice link. But basically the question has two aspects-1. list of conjectures which follow iff the $k$-tuple conjecture is true and 2. list of the results that follows iff the 2nd Hardy-Littlewood conjecture is true. Tao's remark may apply to the 2nd perhaps but not in my opinion to the 1st part. $\endgroup$
    – user57432
    Commented Jan 9, 2015 at 12:21

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