The St. Ives matching conjecture for online path Ramsey numbers

For each $k\geq 1$, let $S_k$ be the St. Ives ordered matching defined recursively from two disjoint consecutive copies of $S_{k-1}$ by adding an edge whose endpoints lie respectively to the left and right of both copies. Let $r_o(S_k,P_n)$ denote the online Ramsey number for finding an ordered copy of $S_k$ or an ordered path $P_n$.

St. Ives matching conjecture. For all $k\geq 1$, there is a constant $c_k$ such that for all $n\geq 1$,

$$r_o(S_k,P_n)\leq n+c_k.$$

Every intersection-free matching on $k$ edges is a subgraph of $S_k$, so this statement is equivalent to the intersection-free matching conjecture. The source gives no resolution of the conjecture.