Green-function divergence conjecture under logarithmic volume growth

Let $M$ be an arbitrary complete connected Riemannian manifold, let $B(o,r)$ denote the geodesic ball centered at $o$, let $\mu$ be the Riemannian measure, let $G(x,o)$ be the Green function with pole at $o$, and let

$$denote the condition$$

\mu(B(o,r))\leq c,r^{\alpha}(\ln r)^{\frac{\alpha-2}{2}}

for some $o\in M$ and all sufficiently large $r$, with $\alpha>2$. **The Green-function divergence conjecture.** If

is satisfied, then, for every $o\in M$,

$$\int_{B(o,1)^c}G(x,o)^{\frac{\alpha}{\alpha-2}}\,d\mu(x)=\infty.$$

This is motivated by comparison with the cited volume-growth nonexistence theorem and the paper's corresponding theorem; it remains open on arbitrary complete connected Riemannian manifolds.