Balko et al.'s minimum Ramsey number conjecture for ordered paths

Let $P_n$ be the path on $n$ vertices, and consider all orderings of its vertices. Balko et al.'s conjecture. Among all orderings of $P_n$, the alternating path has minimum Ramsey number. The conjecture concerns how the ordering of a graph affects its ordered Ramsey number. The source gives examples of orderings of $P_n$ with Ramsey numbers ranging from quadratic to linear, but does not state a resolution of this conjecture.