Sub-Gaussian universality conjecture for generalized patterned random matrices

Let $X_n$ be an $n\times n$ generalized random Toeplitz, circulant, or symmetric circulant matrix with symmetric standard sub-Gaussian entries, meaning mean-zero, variance-one sub-Gaussian entries. Let $p\geq 2$ be a positive integer and set $W_n=\operatorname{Tr}(X_n^p)$. Sub-Gaussian 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 generalized random reverse circulant or Hankel matrix, the same conclusion is conjectured under the additional assumption that $p$ is even. The conjecture extends Gaussian fluctuation results to symmetric sub-Gaussian entries; the motivation is that sub-Gaussian tails yield comparable $\sqrt{\log n}$ operator-norm bounds, although establishing the corresponding total-variation limits remains open.