Nguyen–Scott–Seymour edge-weighting conjecture for graph quasi-isometries

For $L,C\in \mathbb{N}$, let $G$ and $H$ be graphs and let $\varphi$ be an $(L,C)$-quasi-isometry from $G$ to $H$. An edge-weighting function $w\colon E(H)\to\mathbb{N}$ gives $H$ the weighted-graph metric, denoted $(H,w)$.

Nguyen–Scott–Seymour conjecture. For all $L,C\in\mathbb{N}$, there exists $C'\in\mathbb{N}$ such that if $\varphi$ is an $(L,C)$-quasi-isometry from a graph $G$ to a graph $H$, then there is an edge-weighting function $w\colon E(H)\to\mathbb{N}$ such that the same function $\varphi$ is a $(1,C')$-quasi-isometry from $G$ to the weighted graph $(H,w)$.

The conjecture asserts that suitable positive integer edge weights can remove multiplicative distortion while retaining only bounded additive distortion. The paper proves that this general formulation is false by constructing counterexamples, so it is refuted.