Polynomial bound conjecture for edge-ordered Ramsey numbers of degenerate graphs

An edge-ordered graph is a graph whose edges are equipped with a linear ordering. A graph is $d$-degenerate if every induced subgraph has a vertex of degree at most $d$. For an edge-ordered graph $H$, let $r_{\operatorname{edge}}(H)$ be its edge-ordered Ramsey number. Polynomial bound conjecture. If $H$ is an edge-ordered $d$-degenerate graph on $n$ vertices, then

$$r_{\operatorname{edge}}(H)\leq n^{O(d)}.$$

This would improve the paper's existing bound for edge-ordered Ramsey numbers of sparse graphs. The stronger linear analogue is also left unresolved, so this conjecture remains open.