The shrubdepth characterization by paths

Let $\mathcal{C}$ be a class of graphs, let $\mathcal{P}$ denote the class of paths, and let $\sqsupseteq_\mathrm{FO}$ denote the first-order transduction quasiorder. A graph class has bounded shrubdepth when it is first-order interpretable in some fixed finite-height rooted tree class. The shrubdepth characterization by paths. A class $\mathcal{C}$ has bounded shrubdepth if and only if

$$\mathcal{C}\not\sqsupseteq_\mathrm{FO}\mathcal{P}.$$

This conjecture characterizes bounded shrubdepth by excluding the class of paths as a first-order transduction target. The surrounding discussion presents it as one of the difficult structural questions about the first-order transduction quasiorder; its resolution status is not specified in the source.