Stability of Product-Structure Expressions

Prove or refute that there exists a function g from the positive integers to the positive integers such that every $(\mathcal Q_r^\circ,\ell)$-expression constructed in the proof of the transduction lemma is $g(\ell)$-stable.

Background

The main structural theorem produces bounded Qr∘\mathcal Q_r^\circ-clique-width expressions for first-order interpretations of graphs with product structure. The converse theorem requires a stability condition on such expressions. The authors therefore conjecture that the specific expressions constructed in their proof automatically satisfy a stability bound depending only on the number of colors, which would strengthen the characterization of first-order transductions from product-structure classes.

References

There is a function $g: \mathbb{N} \to \mathbb{N}$ such that each $(\ca Q_r\circ, \ell)$-expressions constructed in the proof of \Cref{lem:transductions} is $g(\ell)$-stable.

— Transductions of Graph Classes Admitting Product Structure  (2501.18326 - Hliněný et al., 30 Jan 2025) in Concluding Remarks, Conjecture \ref{conj:expression-stable}