Multivariate CLT if covariance matrix is bounded?

Question

Assuming a sequence of independent but not iid multivariate random variables $x_i$ drawn from distributions whose covariance matrices all share the same bounds (i.e. $C_1 \le C_i \le C_2$ for all i), is this sufficient for the CLT to apply to the sequence of partial sums of $x_i$? My gut says yes based on the accepted answer to this question, which says as long as the sequence of partial averages of covariance matrices converges, this is sufficient for CLT (and in my case I’m making the assumption that boundedness within the same shared bounds implies convergence of the sequence of running means), but is my gut wrong?