The abelian sandpile minimizes average density

Let $G=(V\cup\{s\},E)$ be a finite graph with designated sink vertex $s$, and let $\rho_G(p)$ denote the stationary average density for the $p$-toppling rule, where $p\in[0,1]$ and $p=1$ gives the abelian sandpile model. Minimum attained by abelian sandpiles. For all $p\in[0,1)$,

$$\rho_G(p)>\rho_G(1).$$

This is the weaker statement that the abelian sandpile model has the smallest average density; the supplied text gives no resolution.