Polymath8b project allowed, building on Zhang's 2013 breakthrough, to prove that there are infinitely prime gaps of size less or equal to 600. Under the generalized Elliott-Halberstam conjecture, one can reach the upper bound 6.
My question is: in early August 2016, what is the narrowest interval $I=[a,b]$ such that we know that there are infinitely many prime gaps whose size belongs to $I$?
I'm essentially interested in unconditional results, but those obtained under very plausible conjectures such as (G)EH and or (G)RH can be of some interest too.
Many thanks in advance.