The characterization of good ordered graphs by monotone caterpillars

Let $G^<$ be a connected ordered graph. An ordered graph is good if it has the corresponding ordered Ramsey property used in the paper, and a monotone caterpillar graph is the class characterized in the source by the forbidden ordered subgraphs in Figure 2.

Monotone-caterpillar characterization conjecture. The ordered graph $G^<$ is good if and only if it is a monotone caterpillar graph.

The authors found no counterexamples among the small graphs they exhaustively searched and verified that there are no $4$-good graphs with at most six vertices other than monotone caterpillar graphs. A proof of the characterization remains open.