Asymptotic Gaussianity conjecture for crossings in set partitions

Let $X_n$ and $Y_n$ be the random variables defined in the paper, representing the relevant crossing statistics on set partitions. Asymptotic Gaussianity conjecture. The distributions of $X_n$ and $Y_n$ are asymptotically Gaussian. This conjecture is motivated by numerical evidence; the paper establishes asymptotic Gaussianity for related variables $X_{n,k}$ and $Y_{n,k}$ in specified ranges of $k$, but does not resolve the case of $X_n$ and $Y_n$.