The polynomial-exponent conjecture for ordered Ramsey numbers of path powers

Let $P_n^t$ be the $t$th power of the ordered path on $n$ vertices, and let $K_n$ be the ordered complete graph on $n$ vertices. Write $R_{<}(G,H)$ for the ordered Ramsey number. Polynomial-exponent conjecture. The bound

$$R_{<}(P_n^t,K_n)\leq n^{O(t)}$$

should hold. The paper proves the stronger explicit bound $R_{<}(P_n^t,K_n)\leq 2^{2t-1}n^{t(2t-1)}$, so this open problem is resolved affirmatively in the source.