Multicolour ordered Ramsey conjecture for powers of paths

For positive integers $r$, $t$, and $n$, let $P_n^t$ be the $t$-th power of the path on $n$ vertices, and let $R_<^r(P_n^t)$ denote the $r$-colour ordered Ramsey number of $P_n^t$. Multicolour path-power conjecture. For all $r$, $t$, and $n$,

$$R_<^r(P_n^t)=O_{r,t}\left(n^{O(r)}\right).$$

The preceding theorem only gives a bound of the form $R_<^r(P_n^t)\leq Dn^{Cr\log r}$, so the conjectured removal of the logarithmic factor remains open in the supplied text.