Polynomial -boundedness conjecture for capped graphs

An ordered graph is a graph equipped with a vertex order. An ordered graph $(G,\prec)$ is capped if, for every four vertices $a\prec b\prec c\prec d$, the edges $ac$ and $bd$ imply the edge $ad$. Let $\omega$ denote the clique number and $\chi$ the chromatic number. Capped-graph polynomial bound conjecture. There is a polynomial function $p$ such that every capped graph with clique number $\omega$ has chromatic number at most $p(\omega)$. The paper notes that this would establish polynomial \chi-boundedness for capped graphs and consequently improve the bounds for the related graph classes studied there; it is presented as an expectation rather than a known result.