Quasi $n$-bottlenecking conjecture

Let $G$ be a graph. For $n\in\mathbb{N}$, say that $G$ is coarsely $n$-bottlenecked when it satisfies the coarse $n$-bottlenecking condition, and say that a graph is $n$-edge bottlenecked when it satisfies the corresponding discrete condition.

Quasi $n$-bottlenecking conjecture. If $G$ is coarsely $n$-bottlenecked, then it is quasi-isometric to an $n$-edge bottlenecked graph.

This conjecture seeks a quasi-isometric discrete model for coarsely $n$-bottlenecked graphs. The supplied text presents it as a proposed conjecture, and gives no resolution.