Unit-multiplicative quasi-isometry in the bounded-tree-width decomposition theorem

Determine whether Theorem 1.2 holds with multiplicative quasi-isometry constant L=1; equivalently, establish whether every connected graph admitting a tree-decomposition whose bags are unions of at most k sets of diameter at most r admits a (1,C)-quasi-isometry to a graph of tree-width at most k for an appropriate constant C.

Background

Theorem 1.2 establishes that a connected graph with a tree-decomposition whose bags are unions of at most k sets of diameter at most r admits an (L,C)-quasi-isometry to a graph of tree-width at most k. The authors note that the theorem is not known with multiplicative constant L=1, which would provide a sharper coarse equivalence.