Transduction preservation conjecture for bounded expression-stable clique-width

Let a graph class be a perturbation of a class of bounded expression-stable $\ca P^\circ$-clique-width, where $\ca P^\circ$ denotes the class of reflexive paths. A first-order transduction maps graph classes to graph classes.

Transduction preservation conjecture. The property of being a perturbation of a class of bounded expression-stable $\ca P^\circ$-clique-width is preserved under taking first-order transductions.

This conjecture is presented as a consequence that would follow from a positive answer to the preceding expression-stability conjecture together with the paper's structural results. Its resolution is not given in the supplied text.