A proof I'm writing rests on something I can't prove, probably beyond my knowledge, but it seems right:
For any two primes $p_k, p_l$ (not necessarily consecutive) such that the distance between them $|p_l - p_k| = n$, there exist infinitely many other primes such that the distance between them is also $n$.
I can't figure out a way to show this; I'm guessing it's probably a known result and referring to it would be enough.