Weak convergence of Radon-Nikodým derivatives

I am working at the moment with a sequence of finite measures $nu_n << mu_n$ converging weakly to $nu << mu$. I have seen that if the Radon-Nikodým derivatives $h_n := dfrac{dnu_n}{dmu_n}$ converge pointwise to some $h$, then $h$ is the Radon-Nikodým derivative of $nu$ wrt. $mu$. I am interested of a somehow similar question.

Imagine $h_n$ is uniformly bounded in $n$, e.g. $0leq h_nleq 1$. Suppose also that $mu_n$ is aboslutely continuous wrt. Lebesgue measure with strictly positive density. That means in particular, that $h_n$ is defined Lebsegue-a.e. We thus may define the measure $dLambda_n := h_n(x) dx$. Because of the boundedness, this sequence is tight and we get at least a converging subsequence. Now, under which condition on the sequence of measures do we get that $Lambda_n$ converges to $Lambda$ given by $dLambda = dfrac{dnu}{dmu}(x)dx$?

I cross-reference a question I asked on MO and which is the origin of my question:

The question therein is somewhat different, because we have a specific scenario in which we may reformulate the problem in other ways, but I am hoping that solutions to the problem above may easily be modified to give solutions to the latter.