The best result concerning bounded gaps between primes, whose existence was first proved by the seminal work of Yitang Zhang two years ago, are to my knowledge all of the form $\liminf_{n \rightarrow \infty} \left(p_{n+1} - p_n\right) \leq C$, where the record is currently $C = 246$, according to this source: http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes . These results imply that for at least one value of $k$, $k\leq 123$, that there are infinitely many primes separated by exactly $2k$.

Suppose we can prove that for some positive integer $k$, that there are infinitely many consecutive primes $p_n, p_{n+1}$ such that $p_{n+1} - p_n = 2k$. Can we prove that there are infinitely many consecutive primes separated by exactly $2k+2, 2k+4, \cdots$ etc.? That is, does the existence of infinitely many twin primes for example imply that all even gaps must occur infinitely often? Certainly, whatever technique was used to prove the existence of infinitely many two primes should allow us to just as easily prove it for any even gap, but my question is a bit more specific: namely, can one directly prove the existence of infinitely many primes separated by $2k$ knowing only that there are infinitely many primes separated by $\max\{2, 2k-2\}$?