The stable hereditary graph class characterization conjecture

Let $\mathscr C$ be a hereditary class of graphs, meaning that every induced subgraph of every graph in $\mathscr C$ also belongs to $\mathscr C$. A low shrubdepth decomposition with $n^{o(1)}$ parts is a decomposition of an $n$-vertex graph into a subpolynomial number of parts such that every fixed-size collection of parts induces a graph from a class of bounded shrubdepth. The properties under consideration are first-order transducibility from nowhere dense classes, admitting such decompositions, monadic stability, and stability.

Stable hereditary graph class characterization conjecture. The following properties are equivalent:

1. $\mathscr C$ is a first-order transduction of a nowhere dense class;
2. $\mathscr C$ admits low shrubdepth decompositions with $n^{o(1)}$ parts;
3. $\mathscr C$ is monadically stable;
4. $\mathscr C$ is stable.

This is proposed as a strengthening of the monadic stability characterization conjecture. The source gives no resolution evidence, so the equivalence remains open.