The first Hardy–Littlewood conjecture for prime constellations

Let $0<m_1<m_2<\cdots<m_k$ be integers. Define $P=(p,p+2m_1,p+2m_2,\dots,p+2m_k)$ and let $\pi_P(x)$ denote the number of primes $p\leq x$ for which all of $p,p+2m_1,p+2m_2,\dots,p+2m_k$ are prime. For a prime $q$, let $w(q;2m_1,2m_2,\dots,2m_k)$ be the number of distinct residues of $0,2m_1,2m_2,\dots,2m_k$ modulo $q$. The tuple is admissible when it does not form a complete residue class modulo any prime.

First Hardy–Littlewood conjecture. Unless $P$ forms a complete residue class with respect to some prime, $\pi_P(x)$ is asymptotic to

$$2^k\prod_{\substack{q\text{ prime}\q\geq3}}\left(1-\frac{1}{q}\right)^{-k-1}\left(1-\frac{w(q;2m_1,2m_2,\dots,2m_k)}{q}\right)\int_{2}^{x}\frac{1}{(\log t)^{k+1}}\,dt.$$

This is the prime-constellation generalization of the prime number theorem, predicting the asymptotic number of prime patterns with prescribed even gaps. Its status is not resolved in the supplied source material.