The ordered matching conjecture

An ordered graph is a graph together with a total vertex ordering. The ordered matching conjecture. Let $(M,\prec)$ be an ordered graph with maximum degree $1$. Then the class of $(M,\prec)$-free ordered graphs is $\chi$-bounded. The paper explains that this would imply dichromatic binding for tournaments admitting a backedge graph of maximum degree $1$; the conjecture is attributed to Briański, Davies and Walczak.