Divergence of the expected odometer at criticality in activated random walk

Let $m(0)$ denote the odometer at the origin for fixed-energy activated random walk with i.i.d. initial particle density $\rho$, and let $\rho_c(\lambda)$ be its critical density for sleep rate $\lambda>0$. Expected-odometer divergence conjecture. For any sleep rate $\lambda>0$, as $\rho earrow \rho_c(\lambda)$, we have

$$\mathbf{E}_\rho\bigl[m(0)\bigr] \nearrow \infty.$$

At criticality, fixed-energy activated random walk is generally believed to remain active forever, which would make the odometer infinite at every site almost surely. The stated divergence of the expected odometer is presented as a more approachable open question.