Kozma's rough-isometry conjecture for $L_\infty$ mixing times

Let $G$ and $H$ be finite graphs that are $K$-roughly isometric, and suppose both have maximal degree at most $d$. Denote by $\tau_{\infty}(G)$ and $\tau_{\infty}(H)$ the $L_{\infty}$ mixing times of simple random walk on $G$ and $H$, respectively. Kozma's conjecture. There exists a constant $C(K,d)$ such that

$$\tau_{\infty}(G) \;\leqslant\; C(K,d)\tau_{\infty}(H).$$

The conjecture asserts robustness of $L_{\infty}$ mixing time under rough isometries for bounded-degree finite graphs. The supplied context does not indicate whether it has been proved or disproved.