Asymptotic density of Zhang's primes By this point, it is well known that Yitang Zhang's result implies for some $c$, there are infinitely many primes $p$ such that $p+c$ is also prime, and that the smallest such $c$ is less than $70,000,000$ (which was reduced to $246$ thanks to all who worked on the Polymath Project).  
I'm wondering if anyone has extended this to determine the asymptotic density of these primes, even though we don't know the precise value of $c$.  
Thanks in advance for your comments
 A: Suppose $\mathcal{H}_m$ is an admissible tuple of size $m$, as in the language of Zhang-Tao-Maynard-Polymath. For instance, $(t, t+2, t+6)$ is an admissible $3$-tuple. Then the results of Zhang-Tao-Maynard-Polymath have shown the existence of some $k(m)$, depending only on the size of $m$, such that there are infinitely many $t$ for which admissible $k$-tuples have at least $m$ simultaneous primes.
János Pintz, in his paper Polignac Numbers, Conjectures of Erdős on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded
Gap Conjecture, shows that this $k$ also gives a lower bound on the density, in the sense that there are at least
$$ \Omega\left(\frac{X}{\log^{k(m)}(X)}\right)$$
$k$-tuples with $t \leq X$ with $m$ simultaneous primes. Asymptotically, this means that the density of such $m$-simultaneous primes is at least
$$ \frac{c}{\log^{k(m)}(X)}$$
for some constant $c$.
For example, suppose that $k(3) = 1000$, meaning that admissible $1000$-tuples have $3$ simultaneous primes infinitely often. Then in fact the density of such an infinite configuration (which we do not know) is something like
$$ \frac{1}{\log^{1000}(X)}.$$
Of course, we expect the true density to be much more like
$$ \frac{1}{\log^3 (X)}.$$
