Let $ A $ be a set of odd primes such that between any two consecutive elements thereof there is at least one prime gap that occurs infinitely often, i.e. an even integer $ g $ such that the equality $ p_{n+1}-p_{n}=g $ is true for infinitely many values of $ n $ and let's denote by $ A' $ the set of such $ g $ 's.

Can it be proven with current technology that $ \lim_{x\to\infty}\dfrac{\#\{g\in A',g\leqslant x\}}{x}=\dfrac{1}{2} $? If not what is the best lower bound currently reachable for the considered limit ? I'm mainly interested in unconditional results.

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