A conjecture of De Giorgi on weighted Sobolev spaces 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: On De Giorgi’s conjecture: Recent progress and open problems by Chan and Wei reviews the status of the problem in 2017.
A second attempt to locate this elusive conjecture: De Giorgi gave a talk in Lecce (Italy) in 1995 entitled "Congetture sulla continuità delle soluzioni di equazioni lineari ellittiche autoaggiunte a coefficienti illimitati" (Conjectures on the continuity of solutions of selfadjoint elliptic linear equations with unbounded coefficients"). The talk was typed out but never published.$^\ast$ Progress on these conjectures is discussed in


*

*Xiao Zhong, Discontinuous solutions of linear, degenerate elliptic equations (2008).

*Sungwon Cho, A certain example for a De Giorgi conjecture (2014).


$^\ast$ De Giorgi's Bibliography says: 
"De Giorgi, in particular in the last years of his life, used to
circulated his writings among friends and colleagues, asking for
opinions. We plan in the future to collect and to make available all
these unpublished writings." I tried to contact professor Zhong for a copy, but the email bounced.
 
A: 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
