conjecture: If $\exp(tw), \exp(tw^{-1}) \in L^1_{\operatorname{loc}}$ for each $t > 0$ then $w$ is regular weight.
$w$ is regular if weighted Sobolev space $W^l_p(\Omega,w)$ is equal to the completion of $C^{\infty}$ with respect to the weighted norm $\|\cdot \mid W^l_p(\Omega,w)\|$.
Question: Is there some progress with this conjecture?