Rough-isometry invariance of mixing time for bounded-degree graphs

Let $G$ and $H$ be graphs with $\deg G,\deg H\leq d$, equipped with their path metrics. Let $f:G\to H$ be a rough isometry with constant $K$, meaning that for all vertices $a,b\in G$,

$$\frac{1}{K}d(a,b)-K\leq d(f(a),f(b))\leq Kd(a,b)+K,$$

and every vertex of $H$ lies within distance $K$ of some vertex in the image of $f$. Write $\tau(G)$ and $\tau(H)$ for the mixing times of the simple random walks on the two graphs.

Rough-isometry invariance conjecture. There is a constant $C(K,d)$ such that

$$\tau(G)\leq C(K,d)\tau(H).$$

The claim asks whether mixing time is a geometric property in the setting of unweighted graphs of uniformly bounded degree, where the path metric provides the natural geometry. The supplied text presents it as a formal question and gives no resolution, so its status remains open.