Consider the triangular array $X_{n,k}$ such that, for each $n>0$, the variables $(X_{n,1},\cdots,X_{n,n})$ have the following properties:

- For any given $1 \le L \le n$, all subsets of $(X_{n,1},\cdots,X_{n,n})$ of size $L$ have the same joint distribution (even after applying an arbitrary permutation).
- Each $X_{n,k}$ has zero mean, variance $0<\sigma^2<\infty$, and third absolute moment $0<\rho<\infty$
- Each $(X_{n,k_1},X_{n,k_2})$ pair ($k_1 \ne k_2$) has covariance $\frac{-C}{n}$ for some $0 < C < \infty$ (hence the variables are all negatively correlated, and the correlation tends to zero)

Let $S_n = \sum_{k=1}^{n} X_{n,k}$. Is $\frac{S_n}{\sqrt{\mathrm{Var}[S_n]}}$ asymptotically distributed as $N(0,1)$? If so, is a convergence rate of $O(\frac{1}{\sqrt{n}})$ achieved (cf. Berry-Esseen theorem)? What about in the multivariate setting where each $X_{n,k}$ is a random vector (and the relevant quantities above are replaced by vectors/matrices)?

If it helps, we can also assume that the $X_{n,k}$ are uniformly bounded in $n$ and $k$ with probability one. Answers with further assumptions than the ones listed will also be appreciated.