Cycle–graph Ramsey goodness with linear n

Determine whether there exists a constant c > 0 such that, for all graphs H and all n ≥ c · |H|, the Ramsey number r(C_n, H) equals (n − 1)(χ(H) − 1) + σ(H).

Background

Haslegrave, Hyde, Kim, and Liu proved that r(C_n, H) achieves the lower bound (n−1)(χ(H)−1)+σ(H) when n ≥ c |H| (log χ(H))^4, close to the conjectured linear threshold. Removing the polylogarithmic factor in χ(H) would settle the precise dependence on |H| alone.