The standard conjecture is that the statistics of $(p_{n+1}-p_n)/\ln n$ are, in the limit, precisely a Poissonian distribution with parameter $1$; one can recover conjectures about individual moments from this, although evaluating the moments is the natural way to approach the conjecture. This is the natural heuristic that follows from the model that every integer $m$ is prime with probability $1/\ln m$. I believe this conjecture was proved conditionally by [Gallagher (in Mathematika in 1976)][1], conditionally on a strong form of the prime $k$-tuples conjecture.


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=409385