Preservation of Expression-Stable Clique-Width Under First-Order Transductions

Prove or refute that the property of being a perturbation of a class of bounded expression-stable $\mathcal P^\circ$-clique-width is preserved under first-order transductions.

Background

The paper defines bounded expression-stable $\mathcal H$-clique-width by requiring graph representations through stable parameterized expressions. The authors propose that the resulting property, after allowing bounded perturbations, should be invariant under first-order transductions. A positive answer to the preceding stability conjecture would imply this preservation statement, but the preservation claim is presented separately as an unresolved conjecture.