Very strongly expected to be true. This would follow, for example, from the conjecture that there are infinitely many prime triplets of the form (p, p+6, p+12), and such special cases of Schinzel's hypothesis H are expected to be true. The result of there being infinitely many arithmetic progressions in primes (of any fixed length) is also related - one would only need those primes to be consecutive to get the result here.