The maximal-antichain conjecture for paths and tree classes

Let $\mathcal{P}$ denote the class of paths, let $\mathcal{T}_n$ denote the class of rooted trees of height at most $n$, and let $\sqsupseteq_\mathrm{FO}$ denote the first-order transduction quasiorder. A collection of graph classes is a maximal antichain when its members are pairwise incomparable under this quasiorder and no further class can be added while preserving incomparability. The maximal-antichain conjecture for paths and tree classes. The pairs

$$\{\mathcal{T}_n,\mathcal{P}\}\quad\text{with }n>1$$

are all the maximal antichains containing $\mathcal{P}$. This is presented as an equivalent formulation of the preceding shrubdepth characterization and concerns the structure of antichains in the first-order transduction quasiorder. The source does not specify whether it has been resolved.