Suppose I have some triangular array $\{X_{n,j}\}$ of random variables, which need not be independent or identically distributed. Suppose I further know that $$Var\left(\sum_{j=1}^n X_{n,j}\right)\to \sigma^2 \text{ and } \sum_{j=1}^n \mathbb{E}|X_{n,j}|^{2+\epsilon}\to 0,$$ for some $\epsilon>0$. Then this triangular array satisfies the Lyapunov condition. However, for Lyapunov's CLT we further require independence for each $n$. Suppose instead that we have asymptotic independence in the sense that for any $t\in\mathbb{R},$ $$\left|\mathbb{E}\left[\prod_{j=1}^n\exp\left(itX_{n,j}\right)\right] - \prod_{j=1}^n\mathbb{E}\left[\exp\left(itX_{n,j}\right)\right]\right|\to 0,$$ where $i$ denotes the imaginary unit. Can we then still conclude that $$\sum_{j=1}^n X_{n,j}\to N(0, \sigma^2)?$$

It is easy to see (by a Levy continuity theorem argument) that this is true if the covariances also fade fast enough, so that $$\sum_{j=1}^nVar\left( X_{n,j}\right)\to \sigma^2,$$ but do we need this or can we also conclude asymptotic normality without this additional condition?