Edge-contraction and subdivision conjecture for graph quasi-isometries

Let $\mathcal{G}$ be a class of connected graphs closed under contracting edges and subdividing edges. For graphs $G$ and $H$, an $(L,C)$-quasi-isometry has multiplicative distortion $L$ and additive distortion $C$.

Edge-contraction and subdivision conjecture. For all $L,C\in\mathbb{N}$, there exists $C'\in\mathbb{N}$ such that if a graph $G$ is $(L,C)$-quasi-isometric to a graph in $\mathcal{G}$, then $G$ is $(1,C')$-quasi-isometric to a graph in $\mathcal{G}$.

This is a weakening of the refuted edge-weighting conjecture. It remains open for classes of connected graphs closed under edge contraction and subdivision.