Lopez's path-transduction conjecture for non-2-well-quasi-ordered graph classes

Let $\mathcal{C}$ be a hereditary class of finite graphs. Lopez's path-transduction conjecture. If $\mathcal{C}$ is not $2$-well-quasi-ordered, then $\mathcal{C}$ existentially transduces the class of all finite paths. This conjecture proposes a logical obstruction to $2$-well-quasi-ordering; it remains open in the general setting considered here.