Transduction order of graph classes embeddable in surfaces

A surface is a compact $2$-dimensional manifold without boundary. Write $\Sigma\preccurlyeq\Gamma$ if every graph embeddable in $\Sigma$ is also embeddable in $\Gamma$, and let $\mathscr{E}_\Sigma$ be the class of graphs embeddable in $\Sigma$. Write $\mathscr{A}\sqsubseteq_{\mathsf{FO}}\mathscr{B}$ when $\mathscr{A}$ is first-order transducible from $\mathscr{B}$.

Surface transduction-order conjecture. Let $\Sigma$ and $\Gamma$ be surfaces such that $\Sigma\not\preccurlyeq\Gamma$. Then

$$\mathscr{E}_\Sigma\not\sqsubseteq_{\mathsf{FO}}\mathscr{E}_\Gamma.$$

This conjecture asserts that, among graph classes embeddable in fixed surfaces, the obvious inclusions are the only transducibility relations. It is not known even for the torus and the sphere, where it asks whether toroidal graphs are first-order transducible from planar graphs.