Following https://math.stackexchange.com/questions/3865486/ascendant-gap-chain-of-a-prime-constellation and Green-Tao theorem for 1-central numbers, say two prime constellations of length $k$ are conjugate if they share the same ascendant gap chain. My idea is that one may see the set of all prime constellations of length $k$ that are conjugate to a given one among the set of all prime constellations of length $k$ as an analogue of a conjugacy class $C$ among a group $G$ and hopefully use (the analogue of) Chebotarev density theorem to prove that all conjugate prime constellations of a given length appear asymptotically the same number of times. Indeed the prime gaps appearing in a given prime constellation can be seen as analogues of roots of a split polynomial of degree $k-1$ and permutations thereof as maps sending a prime constellation to conjugates thereof.

Would this, if such an idea works, together with Pintz' theorem quoted by GH from MO in the second link, get us any closer to Hardy-Littlewood $k$-tuple conjecture? If yes, assuming it were not sufficient to prove it, what would be the missing element to achieve a full resolution of this conjecture?

Edit: it seems that a positive answer to my question on MSE would lead to the right notion of the analogue of a conjugacy class would be the set of permutations of prime gaps preserving a given constellation, and that the analogue of the Galois group would be the set of permutations of positions of $k$ primes preserving the admissibility of the $k$-tuple forming a prime constellation. That way, the analogue of the Chebotarev theorem would be that each prime constellation of length $k$ appears with a fixed asymptotic frequency, namely $f_{k}=\frac{1}{N_{k}}$ where $N_{k}$ is the number of distinct prime constellations of length $k$.