Convergence in distribution implies convergence in density

Question

Let $\{F_n\}$ be a sequence of absolutely continuous (wrt Lebesgue measure $\mu$) distributions on $R^k$, and $\{f_n\}$ be a corresponding sequence of densities. Suppose $F_n \Rightarrow F$, where $\Rightarrow$ stands for weak convergence. Can we show that $f_n$ converges pointwise to $f$? Here $f$ is the corresponding density of $F$. I know, in general, this is not true if no further assumptions are imposed (see e.g. Sweeting, 1986). How about $F$ is the normal distribution with zero mean and some positive definite covariance matrix $\Sigma$?