Asymptotic order conjecture for twins in ordered $r$-matchings

An ordered $r$-matching is an ordered matching in which every vertex belongs to one of $r$ edge labels, and let $t^{(r)}(n)$ be the minimum, over ordered $r$-matchings of size $n$, of the maximum size of a pair of twins. Asymptotic order conjecture for twins in ordered $r$-matchings. For every $r\geqslant 2$,

$$t^{(r)}(n)=\Theta\left(n^{\frac{2}{r+1}}\right).$$

This conjecture proposes the true asymptotic order of the extremal function, building on the paper's probabilistic upper bound and earlier results on twins in permutations. The source presents it as a problem for future consideration and gives no resolution.