Stable transduction conjecture for hereditary graph classes

Let $\mathscr C$ be a hereditary class of graphs. A class is stable if no transduction of it contains all finite half-graphs, and it is nowhere dense in the standard sparse graph-theoretic sense. A transduction is a first-order interpretation operation between classes of structures.

Stable transduction conjecture. A hereditary class of graphs is stable if and only if it is a transduction of a nowhere dense class of graphs.

The conjecture seeks a normal form for stable hereditary classes in terms of sparse classes. The supplied text says that it has been proposed as a characterization and gives no evidence of resolution.