Equality of inner and outer densities for activated random walkers

Fix a sleep rate $0<\lambda\leq eq\infty$. Let $(\eta(x))_{x\in\mathbb Z}$ be independent with $\mathbb P(\eta(x)=1)=\zeta$ and $\mathbb P(\eta(x)=0)=1-\zeta$, and let $w$ be the $\mathrm{ARW}^{\lambda}$ odometer. Define the outer density by

$$\zeta_{out}(\mathbb Z,\lambda)=\sup\{\zeta>0:\mathbb E_\zeta(w(0)^3)<\infty\}.$$

For an interval $I\subseteq\mathbb Z$, let $\mathcal S_I(\mathbf 1_I)$ be the stabilization of the all-active configuration with a sink at the boundary, and define

$$\zeta_{in,I}(\mathbb Z,\lambda)=\inf\{\zeta>0:\mathbb P(|\mathcal S_I(\mathbf 1_I)|>\zeta\# I)\leq(\# I)^{-20}\},\qquad \zeta_{in}=\limsup_I\zeta_{in,I}.$$

Density equality conjecture.

$$\zeta_{in}=\zeta_{out}.$$

The paper proves the inequality $\zeta_{in}\geq\zeta_{out}$. Equality would, together with the main theorem, imply that the single-source visited set is asymptotically a centered interval. The source gives no evidence of a resolution.