Hardy–Littlewood's asymptotic density conjecture for admissible prime tuples

Let $(a_1,a_2,\ldots,a_k)$ be a monotonically increasing sequence of positive even integers. Let $p$ be prime, and let $P=(p,p+a_1,p+a_2,\ldots,p+a_k)$ be an admissible prime $k$-tuple, meaning that the entries are prime and do not form a complete residue class modulo any prime. Let $\pi_P(n)$ count primes $p<n$ for which every $p+a_i$ is prime. Let $w(q;a_1,a_2,\ldots,a_k)$ be the number of distinct residues of $a_1,a_2,\ldots,a_k$ modulo the prime $q$. Hardy–Littlewood's asymptotic density conjecture. The counting function satisfies

$$\pi_P(n) \sim C_{a_1,a_2,\ldots,a_k}\int_2^n \frac{dt}{\log^{k+1}t},$$

where

$$C_{a_1,a_2,\ldots,a_k}=2^k\prod_q \frac{1- \frac{w(q;a_1,a_2,\ldots,a_k)}{q}}{(1-\frac 1 q)^{k+1}}.$$

This is the Hardy–Littlewood prediction for the asymptotic density of admissible prime $k$-tuples; if true, it implies the infinitude of every admissible prime tuple. The paper provides numerical data supporting the conjecture, but it remains unproved in general.