Bounded-tree-width target for quasi-isometries to tree-width two

Prove that for all L,C there exist C' and k such that every graph admitting an (L,C)-quasi-isometry to a graph of tree-width at most two admits a (1,C')-quasi-isometry to a graph of tree-width at most k.

Background

The general class statement in Conjecture 1.4 is known for tree-width one and for outerplanar graphs, a special subclass of the graphs of tree-width two. The authors state that the corresponding result for arbitrary graphs of tree-width two remains open and formulate Conjecture 1.8 as a weaker version, allowing the resulting target to have some bounded tree-width k rather than requiring tree-width two.

References

But for tree-width two in general, the result is open, as is the following weaker statement: 1.8. Conjecture: For all L, C there exist C', k such that if there is an (L, C)-quasi-isometry from a graph G to a graph of tree-width at most two, then there is a (1, C')-quasi-isometry from G to a graph of tree-width at most k.

— Coarse tree-width  (2501.09839 - Nguyen et al., 16 Jan 2025) in Conjecture 1.8, Section 1