Near-quadratic ordered Ramsey numbers for bounded-bandwidth graphs

Establish whether R_<(P^<_{k,n}) = n^{2+o(1)} as n tends to infinity for every fixed positive integer k.

Background

For the ordered graph P^<_{k,n}, the survey reports an upper bound of order n^{4+o(1)} and only a quadratic lower bound. The cited authors conjecture that the true growth is close to the lower bound.