Stability conjecture for expressions constructed from transductions

Let $\ca Q_r^\circ$ denote the class of reflexive $r$-ary paths, let $\ell\in\mathbb{N}$, and consider the $(\ca Q_r^\circ,\ell)$-expressions constructed in the proof of the transduction lemma. A function $g:\mathbb{N}\to\mathbb{N}$ should exist such that every such expression is $g(\ell)$-stable.

Expression-stability conjecture. There is a function $g:\mathbb{N}\to\mathbb{N}$ such that each $(\ca Q_r^\circ,\ell)$-expression constructed in the proof of the transduction lemma is $g(\ell)$-stable.

The conjecture concerns the stability of the expressions arising when first-order transductions are applied to monadically stable graph classes admitting a product structure. A positive answer would combine with the cited structural theorems to characterize classes first-order transducible from classes admitting a product structure as perturbations of classes of bounded expression-stable reflexive-path clique-width.