It is known that the weak convergence of joint distribution does not imply the weak convergence of conditional distribution (for example, see this post). What happens if we assume that the density functions always exist? I have a simple proof but I am not sure if it is correct.
Suppose $(X_n, Y_n) \stackrel{d}{\to} (X, Y) \in \mathbb{R} \times \mathbb{R}$. Assuming that CDFs are continuous and density functions $p_{X_n, Y_n}$ exist, we can write the conditional density function as: \begin{align} p_{Y_n|X_n}(y|x) = \frac{p_{X_n, Y_n}(x, y)}{\int_{\mathbb{R}}p_{X_n, Y_n}(x, y') dy'} \label{eq} \tag{1} \end{align} Weak convergence is equivalent to the convergence of CDFs in their continuity points and the density functions (?). Now, both the denominator and numerator of eq. (\ref{eq}) converge, therefore the conditional density converges.
I am not sure if the convergence of the CDFs results in the convergence of density functions. If that is not true in general, are there any set of general assumptions that can guarantee that?
Thank you!