Nonexistence conjecture for superlinear elliptic inequalities on weighted graphs

Let $(V,\b5)$ be a weighted graph, let $o\in V$, and let $\sigma>1$. A nonnegative solution $u$ of the elliptic inequality

$$\Delta u+u^{\sigma}\leq 0$$

is a function on $V$ satisfying this inequality, where $B(o,n)$ denotes the ball of radius $n$ centered at $o$ and $\mu(B(o,n))$ its weighted volume. Nonexistence conjecture. If

$$\sum_{n=1}^{\infty}\frac{n^{2\sigma-1}}{\mu(B(o,n))^{\sigma-1}}=\infty,$$

then the only nonnegative solution to the elliptic inequality is identically zero. This would sharpen the stated nonexistence theorem by removing condition $(p_0)$, whose necessity is not known.