Let $\mu$ be a probability measure on $\mathbb{R}^d$ which is absolutely continuous with respect to the Lebesgue measure with density $\rho$. Assume that, for all $t>0$, 
\begin{align\*}
\exp \left(t \rho\right), \exp \left(t \rho^{-1}\right) \in L^1\_{loc}.
\end{align\*}
Then, a conjecture of De Giorgi asserts that the Meyers-Serrin theorem holds for the weighted Sobolev spaces associated with $\mu$, namely $H=W$.
The only reference I know speaking of this conjecture is this paper 


<https://iopscience.iop.org/article/10.1070/SM1998v189n08ABEH000344/meta>


Thus, I would like to know if there have been recent advances in proving or disproving this conjecture.


Thanks in advance.



A conjecture of De Giorgi on weighted Sobolev spaces
I did some diggings and some readings and found out that the conjecture has been solved here 


<https://link.springer.com/article/10.1134/S1064562413060173>


and extended recently to a wider context in 


<https://www.degruyter.com/view/journals/crll/2019/746/article-p39.xml>


