Cutoff for Activated Random Walk at the stationary density

Let $u$ be the uniform driving sequence on $B_r = B(0,r) \cap \mathbb{Z}^d$. Let $0 < \lambda < \infty$ be a constant sleep rate, and let $P$ be the simple random walk on $B_r$ with sink at $\mathbb{Z}^d \setminus B_r$. Cutoff for Activated Random Walk at the stationary density. There exists a constant $\zeta = \zeta(\lambda,d)<1$ such that, for every $\epsilon>0$,

$$\frac{t_{\mathrm{mix}}({\tt ARW},u,\epsilon)}{\# B_r} \to \zeta \qquad \text{as } r \to \infty,$$

and, writing $|{\tt S}[1_{B_r}]|$ for the number of particles in the Activated Random Walk stationary state on $B_r$,

$$\frac{|{\tt S}[1_{B_r}]|}{\# B_r} \to \zeta \qquad \text{in probability as } r \to \infty.$$

Moreover, $\zeta = \zeta_c$, the critical density for Activated Random Walk stabilization in $\mathbb{Z}^d$. This conjecture relates mixing-time cutoff and stationary particle density to the critical density for Activated Random Walk stabilization; the source states it without resolving any of the three assertions.