The pathwidth–treewidth incomparability conjecture

For each positive integer $n$, let $_n$ be the class of graphs of pathwidth at most $n$, and let $_n$ be the class of graphs of treewidth at most $n$. Two graph classes are non-comparable if neither FO-transduces the other. The pathwidth–treewidth incomparability conjecture.

$$\mathcal{PW}_{n+1}\text{ and }\mathcal{TW}_n\text{ are non-comparable.}$$

The authors state that this would imply strictness of the treewidth hierarchy, analogous to their proved strictness result for pathwidth. The supplied text does not establish a resolution.