Universality conjecture for generalized patterned Gaussian random matrices

Let $X_n$) be an $n\times n$ generalized random Toeplitz, circulant, or symmetric circulant matrix with standard Gaussian entries. Let $p\geq 2$ be a positive integer and set $W_n=\operatorname{Tr}(X_n^p)$. Universality conjecture. As $n\to\infty$, the normalized statistic

$$\frac{W_n-\mathbb{E}(W_n)}{\sqrt{\operatorname{Var}(W_n)}}$$

converges in total variation to $N(0,1)$. For a reverse circulant or Hankel matrix with correlated entries, the same conclusion is conjectured under the additional assumption that $p$ is even. This would remove the covariance-decay restrictions in the paper's Gaussian fluctuation results; the authors explain that proving it likely requires sharper simultaneous bounds or a new method, and note possible phase transitions for odd powers in the reverse circulant and Hankel cases.