Recall that for a domain $\Omega\subset \mathbb{R}^n$, the weighted Sobolev space $W^{1,n}(\Omega,\mu)$ is defined as $f\in L^n(\Omega,\mu)$ and the weak derivative $Df\in L^n(\Omega,\mu)$.

Let now $d\mu(x)=w(x)dx$, where $w:\mathbb{R}^n\to [0,1]$ is a weight. We also assume that $w^{-\frac{1}{n-1}}\in L^1(\Omega,dx)$, where $dx$ represent the Lebesgue measure. I have a continuous function $f\in W^{1,1}(\Omega,dx)$ and I know that $Df\in L^n(\Omega,\mu)$ (so in particular $f\in W^{1,n}(\Omega,\mu)$ as well). I wonder is it ture that there exists a sequence of Lipschitz continuos functions $f_i$ such that $f_i\to f$ locally uniformly and that $|Df_i|\to |Df|$ in $L^n(\Omega)$?

I know the answer is positive if $\mu$ is an $A_p$-weight. I wonder any known weaker condition ensures the above conclusion. Thanks for all possible references on this result.