The conjectured order of twins in ordered r-matchings

For each integer $r\geqslant2$, let $t_r^{\operatorname{match}}(n)$ denote the largest size of twins that are unavoidable in every ordered $r$-matching of size $n$. Here twins are pairs of disjoint, order-isomorphic sub-matchings. Twins conjecture for ordered r-matchings. For every fixed $r\geqslant 2$, one has

$$t_r^{\operatorname{match}}(n)=\Theta\left(n^{\frac{r}{2r-1}}\right).$$

This extends the paper's results and proposed asymptotic order for twins beyond ordinary ordered matchings. Its status is open in the supplied text.