The weighted Sobolev decay conjecture for the Green kernel of the p-Laplacian

Let $M$ be a metric-measure space, let $u>p>1$, and let $\Omega\subset M$ be a connected open set containing a fixed origin $o$. Write $r(x)=\operatorname{dist}(x,o)$, and let $\mathcal{G}$ denote the Green kernel of the $p$-Laplacian on $\Omega$ with pole at $o$. Assume that $\Omega$ supports the weighted Sobolev inequality

$$\left( \int_{\Omega} \eta(r)^{-\frac{p}{\nu-p}} |\psi|^{\frac{\nu p}{\nu-p}}\right)^{\frac{\nu-p}{\nu}} \le \mathscr{S}_{p,\nu} \int_{\Omega} |\nabla \psi|^p \qquad \forall\,\psi\in\operatorname{Lip}_c(\Omega),$$

where $\mathscr{S}_{p,\nu}>0$ and $\eta\in C(\mathbb{R}^+_0)$ is positive and non-decreasing. The weighted Sobolev decay conjecture. The $p$-Laplacian is non-parabolic on $\Omega$ and

$$\mathcal{G}(x)\le C_{p,\nu}^{\frac{1}{p-1}}\eta\bigl(2r(x)\bigr)^{\frac{1}{p-1}}r(x)^{-\frac{\nu-p}{p-1}},\qquad \forall x\in\Omega\setminus\{o\},$$

where

$$C_{p,\nu}=\mathscr{S}_{p,\nu}^{\frac{\nu}{p}}\left[2^{\nu}p(1+p)^p\left(\frac{p}{p-1}\right)^{p-1}\right]^{\frac{\nu-p}{p}}.$$

Moreover, $C_{p,\nu}$ is bounded as $p\to1$ whenever $\mathscr{S}_{p,\nu}$ is, and if $\eta(t)=o(t^{\nu-p})$ as $t\to\infty$, then $\mathcal{G}(x)\to0$ as $r(x)\to\infty$ in $\Omega$. The conjecture concerns decay and non-parabolicity of the Green kernel under a weighted Sobolev inequality; the source attributes it to earlier work, but the supplied text gives no resolution, so its status remains open.