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.