FO transductions between surface-embeddable graph classes

Prove that for surfaces $\Sigma$ and $\Gamma$ with $\Sigma\not\preceq\Gamma$, the class of graphs embeddable in $\Sigma$ is not FO-transducible from the class of graphs embeddable in $\Gamma$.

Background

The survey observes that inclusions between surface-embeddable graph classes yield the obvious transducibility relations. It conjectures that there are no additional FO-transduction relations between these classes, a question that is already open for toroidal graphs versus planar graphs.

References

We conjecture that no more transducibility relations are present between those classes.

— Graph classes through the lens of logic  (2501.04166 - Pilipczuk, 7 Jan 2025) in Conjecture 6.1, Section 6, paragraph “Fine understanding of the transduction order”