Lower bound for a prime gap occurring infinitely often

In his striking paper of may 2013, Zhang showed the existence of an even integer $g\lt 70,000,000$ such that $g$ is a prime gap occurring infinitely often. What is the best unconditional lower bound for the number of primes $p$ below $x$ such that $p'-p=g$, with $p'$ the smallest prime strictly greater than $p$?
Zhang's proof can be refined to show that the number of those primes is $\gg x/\log^k x$, where $k$ is the size of the tuple whose translates contain the relevant pairs, i.e. $k=3{,}500{,}000$ in the original proof and $k=50$ in the current record by PolyMath8b. For more details see the Main Theorem in Pintz's article here.
• So could we deduce from the twin prime conjecture that the number of twin primes below $x$ is $\gg x/log^{2} x$? – Sylvain JULIEN Jul 23 '14 at 17:22
• He didn't say it followed from any conjecture: he said it followed from Zhang's proof. When someone proves the twin prime conjecture, then it will be time to see what the proof implies about the number of twin primes below $x$. – Gerry Myerson Jul 24 '14 at 0:43