Lower-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, and let $n$ denote the number of vertices of $H$. Lower-bound conjecture. For every $d\geq 1$, there exists an infinite family of edge-ordered $d$-degenerate graphs $H$ such that

$$r_{\operatorname{edge}}(H)\geq n^{\Omega(d)}.$$

The authors conjecture that such examples exist because they cannot rule out the stronger linear bound, and that this lower bound would make the polynomial upper bound tight up to the constant in the exponent. This conjecture remains open.