Disclaimer: this question was first asked on Mathoverflow and downvoted there as it doesn't seem to suit it. I hope it is nevertheless sufficiently interesting to be posted here.
This question is a follow-up to https://mathoverflow.net/questions/163191/isometry-group-of-an-integer-as-of-the-corresponding-omegan-parallelotope?r=SearchResults whose content I can copy-paste if some people want it to be self-contained.
Given a prime constellation $C=(0, a_{1},a_{1}+a_{2},\cdots,a_{1}+\cdots+a_{k})$, one can associate to it a group defined the same way as in the link above where this time the $a_{i}$ are prime gaps and not exponents of primes in the prime factorization of an integer. Of course, this requirement prevents many such isometry groups from existing. But suppose $G$ is an admissible "prime constellation isometry group" (PCI group for short), does the fact that $\pi_{G}(x)\to\infty$ in the sense of the link imply that the corresponding prime constellation occurs infinitely often? If yes, can one deduce Hardy-Littlewood conjecture from the relevant asymptotics after sieving out the non admissible PCI groups?