Characterization of finite pairs of chi-unavoidable connected ordered graphs

Let $H$ and $H'$ be $\chi$-unavoidable connected ordered graphs, each with at least two edges. Connected ordered-graph finiteness conjecture. The pair $(H,H')$ is Ramsey finite if and only if $(H,H')$ is either a pair consisting of a right star and an almost increasing right caterpillar, or a pair consisting of a left star and an almost increasing left caterpillar. The conjecture concerns the remaining connected $\chi$-unavoidable cases identified in the paper; the source reports a proof of one small case in a dissertation but leaves the general characterization open.