Optimality conjecture for shifted IDLA mixing time on quasi-regular graphs

Let $G$ be a quasi-regular graph satisfying Assumption and let $t_\gamma$ be the time scale from the mixing theorem, namely the quantity appearing in Theorem.

Optimality conjecture. Under Assumption, $t_\gamma$ is optimal up to logarithmic factors.

This conjecture asserts that the bound for shifted internal diffusion-limited aggregation to forget its initial state cannot generally be improved by more than logarithmic factors. It is motivated by the corresponding near-optimal result for the base graph $G=\mathbb{Z}_N$, while the general quasi-regular case remains open.