Nonexistence conjecture for nonnegative solutions under integral volume conditions

Let $M$ be a noncompact geodesically complete manifold. For $r>0$, let $V(r)$ denote the volume quantity appearing in the integral conditions

$$\int^{\infty}\frac{r^{2p-1}}{V(r)^{p-1}}\,dr<\infty$$

or, equivalently,

$$\int^{\infty}\left[\int_r^{\infty}\frac{t\,dt}{V(t)}\right]^{p-1}r\,dr<\infty.$$

Nonexistence conjecture. If either of these integral conditions is satisfied on $M$, then every nonnegative solution of the equation referred to as

$$is identically zero. The conjecture asks whether the volume-doubling and Poincaré assumptions used in the preceding existence/nonexistence result can be removed. Its precise scope depends on the equation$$

and the definition of $V(r)$ in the paper; the supplied statement does not resolve its status.