Stability characterization for bounded-linear-rankwidth graph classes

Let $\mathscr{C}$ be a class of graphs with bounded linear rankwidth. A first-order transduction is a graph interpretation obtained using first-order formulas. The class $\mathscr{C}$ is stable if no first-order formula defines arbitrarily large orders. Linear-rankwidth stability conjecture. The class $\mathscr{C}$ is included in a first-order transduction of a class $\mathscr{D}$ with bounded pathwidth if and only if $\mathscr{C}$ is stable.

This conjecture is the linear analogue of the rankwidth characterization, relating stability to first-order descriptions by classes of bounded pathwidth. The supplied text gives no resolution status.