Necessity of the tree-width increase in the converse construction

Determine whether the increase from tree-width k to tree-width k+1 is essential when converting a quasi-isometry to a graph of tree-width at most k into a tree-decomposition with bags that are unions of boundedly many bounded-diameter sets and then applying Theorem 1.2.

Background

The converse result produces a tree-decomposition whose bags are unions of at most k+1 bounded-diameter sets when the original target graph has tree-width at most k. Applying Theorem 1.2 then yields a target of tree-width at most k+1 rather than k. The authors explicitly question whether this loss of one in the tree-width bound reflects a genuine obstruction or merely the proof method.