Double clustering conjecture for bounded-doubling graph families

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two families of graphs with bounded doubling dimension, not necessarily with the same constants. For any two graphs $\G_1\in\mathcal{F}_1$ and $\G_2\in\mathcal{F}_2$ of size $n$, construct the double clustering graph using the random-permutation construction: $(u,v)$ is an edge whenever, for every $w\neq u,v$, $d_1(u,w)<d_1(u,v)$ implies $d_2(\pi(u),\pi(w))\geq d_2(\pi(u),\pi(v))$. Bounded-doubling navigability conjecture. The resulting double clustering graph allows greedy routing in $O(\log n)$ expected steps. Questions such as connectivity, diameter, and edge length remain open in some or all cases, so the conjecture is unresolved.