Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question is thus: which even positive integers are known to be both prime gaps (at least once) and the sum of 2 primes? Can we prove that the set of such integers has asymptotic density $1$ among all even positive integers?
Thanks in advance.