The nested matching ordered Ramsey number conjecture

Let $NM_k$ be the ordered matching on $[2k]$ whose edges are $(i,j)$ with $i+j=2k+1$, and let $r_<(G,H)$ denote the off-diagonal ordered Ramsey number. Nested matching conjecture. For any positive integer $k$,

$$r_<(NM_k,K_3)=4k-1.$$

The preceding bounds show that $4k-2<r_<(NM_k,K_3)\leq 6k$, while this conjecture predicts the optimal value up to the exact lower-bound obstruction.