The coarse Menger-type conjecture for asymptotic minors

A graph is coarsely bottlenecked if it satisfies the coarse bottlenecking condition introduced in the paper. For each $n\in\mathbb{N}$, let $D_n$ denote the finite graph used in the paper's asymptotic-minor characterization.

Coarse Menger-type conjecture. If a graph is not coarsely bottlenecked, it must contain $D_n$ as an asymptotic minor for every $n\in\mathbb{N}$.

This conjecture would remove the coarse-bottlenecking hypothesis from the paper's coarse Menger-type characterization. The paper proves the corresponding implication for coarsely bottlenecked graphs, while the unrestricted statement remains open.