The A-infinity weighted Sobolev conjecture

Let $n>1$, let $w$ be a weight in $\textup{A}_\infty$, and let $\mathcal W_1$ denote the class of weights for which the endpoint weighted Sobolev inequality holds for every $u\in\operatorname{Lip}_c(\mathbb R^n)$:

$$\|u\|_{\textup{L}^{\frac{n}{n-1}}(w)}\leq C\|\nabla u\|_{\textup{L}^1(w^{1-\frac1n})}.$$

The $\textup{A}_\infty$ weighted Sobolev conjecture. One has

$$\textup{A}_\infty\subseteq\mathcal W_1,$$

that is, every $w\in\textup{A}_\infty$ satisfies the endpoint weighted Sobolev inequality. The preceding theorem shows that the endpoint case implies the corresponding inequalities for $1<p<n$; the asserted inclusion of all $\textup{A}_\infty$ weights in $\mathcal W_1$ remains conjectural in the source.