Weighted-target improvement for arbitrary quasi-isometries

Establish whether every (L,C)-quasi-isometry from a graph G to a graph H can be converted, using a non-negative integer edge-weight function on H, into a (1,C')-quasi-isometry given by the same vertex map.

Background

Theorem 1.6 proves this weighted-target improvement when H has bounded path-width. Conjecture 1.7 proposes that the path-width assumption may be unnecessary: the target graph itself could retain its combinatorial structure while edge weights absorb the distortion of the original quasi-isometry. The authors state that they believe the conjecture is likely too strong but do not have a counterexample.