The dense analogue characterization conjecture for merge-width and flip-width

Let $\mathcal C$ be a graph class. Following the paper, say that $\mathcal C$ is in the dense analogue of bounded expansion if, for every weakly sparse graph class $\mathcal D$ such that $\mathcal C$ transduces $\mathcal D$, the class $\mathcal D$ has bounded expansion. Dense analogue characterization conjecture. The following conditions are equivalent:

$$\begin{array}{ll} \text{(i)} & \mathcal C\text{ has bounded merge-width},\\ \text{(ii)} & \mathcal C\text{ has bounded flip-width},\\ \text{(iii)} & \mathcal C\text{ is in the dense analogue of bounded expansion}. \end{array}$$

The conjecture would provide an obstruction-based characterization of bounded merge-width and bounded flip-width. It is known in the paper for graph classes excluding a biclique as a subgraph and for classes of ordered graphs, while the general equivalence remains open.