Monotonicity of the average density for stochastic sandpiles

Let $G=(V\cup\{s\},E)$ be a finite graph with designated sink vertex $s$. Under the $p$-toppling rule, each edge incident to a toppled vertex independently sends a particle with probability $p$, and particles sent to $s$ are removed. Let $\mu_G^{(p)}$ be the stationary distribution, and define the average density by

$$\rho_G(p):=\mathbb{E}_p\bigg[\sum_{v\in V}\frac{\eta(v)}{|V|}\bigg].$$

Here $p\in[0,1]$. Monotonicity of the average density. For all $p_1,p_2\in[0,1]$ with $p_1<p_2$,

$$\rho_G(p_1)>\rho_G(p_2),$$

so the average density is strictly monotonically decreasing in $p$. The conjecture is motivated by the observed higher density for uniform topplings on more complicated graphs; the supplied text gives no proof or resolution.