Subquadratic ordered Ramsey bound for matchings of interval chromatic number two

Let $M^<$ be an ordered matching on $n$ vertices, let $K^<_3$ be the ordered triangle, and let $\chi_<(M^<)$ denote the interval chromatic number of $M^<$. Let $r_<(M^<,K^<_3)$ be the ordered Ramsey number of $M^<$ versus $K^<_3$.

Interval-chromatic-two conjecture. There exists an $\epsilon>0$ such that, for every ordered matching $M^<$ on $n$ vertices with $\chi_<(M^<)=2$,

$$r_<(M^<,K^<_3)\in O(n^{2-\varepsilon}).$$

This is a weaker uniform version of the preceding conjecture for interval chromatic number two. The growth rate of $r_<(M^<,K^<_3)$ is not understood even in this case, so the asserted subquadratic bound remains open.