The almost bounded merge-width characterization conjecture

Let $\mathcal C$ be a hereditary graph class. It has almost bounded merge-width if, for each fixed $r\in\mathbb N$, every $n$-vertex graph in $\mathcal C$ has radius-$r$ merge-width at most $n^{o(1)}$; analogously define almost bounded flip-width by replacing merge-width with flip-width. The class $\mathcal C$ is monadically dependent if and only if it does not transduce the class of all graphs. Almost bounded merge-width characterization conjecture. The following conditions are equivalent:

$$\begin{array}{ll} \text{(i)} & \mathcal C\text{ has almost bounded merge-width},\\ \text{(ii)} & \mathcal C\text{ has almost bounded flip-width},\\ \text{(iii)} & \mathcal C\text{ is monadically dependent}. \end{array}$$

For hereditary weakly sparse classes, the first two conditions are equivalent to nowhere denseness, and nowhere dense classes have almost bounded merge-width. The general hereditary characterization by monadic dependence remains open.