2 corrected spelling of Hensley

We say that a $k$-tuple of integers $b_1, \cdots, b_k$ is admissible if for every prime $p$ there exists an integer $X$ such that none of the numbers $X + b_1, \cdots, X + b_k$ is divisible by $p$. Then the prime $k$-tuple conjecture can be stated as follows:

If $b_1, \cdots, b_k$ is an admissible $k$-tuple, then there exists infinitely many values of $X$ such that $X+b_1, \cdots, X+b_k$ are all prime.

Richards and Henley Hensley has shown in 1974 that this conjecture is incompatible with another conjecture due to Hardy and Littlewood, which asserts that $\pi(M+N) \leq \pi(M) + \pi(N)$ for integers $M,N > 1$.

So my question is has either of these conjectures been confirmed to be true or false? Obviously if one of them is true then the other is automatically false. If a counter example exists for either problem, can anyone point out the counter example?

1

# On the prime $k$-tuple problem

We say that a $k$-tuple of integers $b_1, \cdots, b_k$ is admissible if for every prime $p$ there exists an integer $X$ such that none of the numbers $X + b_1, \cdots, X + b_k$ is divisible by $p$. Then the prime $k$-tuple conjecture can be stated as follows:

If $b_1, \cdots, b_k$ is an admissible $k$-tuple, then there exists infinitely many values of $X$ such that $X+b_1, \cdots, X+b_k$ are all prime.

Richards and Henley has shown in 1974 that this conjecture is incompatible with another conjecture due to Hardy and Littlewood, which asserts that $\pi(M+N) \leq \pi(M) + \pi(N)$ for integers $M,N > 1$.

So my question is has either of these conjectures been confirmed to be true or false? Obviously if one of them is true then the other is automatically false. If a counter example exists for either problem, can anyone point out the counter example?