Given some integer $k>0$, there are $O(x/\log^2 x)$ primes $p \le x$ such that $p+2k$ is also prime. It has been conjectured at least since Hardy-Littlewood that $$ \pi_{2k}(x) \sim c_{2k}\int_2^x\frac{dt}{\log^2t} $$ with $$ c_{2k}=2C_2\prod_{p|k,p>2}\frac{p-1}{p-2} $$ where $\pi_{2k}(x)$ is the count of such primes and $C_2$ is the twin prime constant (A005597). i'm interested in the upper bound part.
I've seen a number of papers giving explicit constants $c$ such that $\pi_2(x) \le c\int_2^xdt/\log^2t$ for large enough $x$. What are the best known constants for $\pi_{2k}$? I assume that they're no closer to the (conjectured) optimal constants than our best bounds for $\pi_2$ (Wu [1] is within about 3.4).
I hope there has been some paper making this explicit? It would be fantastic to have a reference proving an upper bound of, say, $10c_{2k}$, but I'm open to whatever can be found. (Perhaps some paper has an upper bound which is unbounded as $k\to\infty$ but finite for all $k$.)
[1] J Wu, Chen's double sieve, Goldbach's conjecture, and the twin prime problem, Acta Arithmetica 114:3 (2004), pp. 215–273. arXiv:0705.1652 [math.NT]
