Cyclic Ramsey conjecture for monotone paths

Let $P_a^\mathrm{mon}$ and $P_b^\mathrm{mon}$ denote monotone paths of orders $a$ and $b$, respectively, and let $R_\mathrm{cyc}$ be the cyclic Ramsey number. Cyclic monotone-path conjecture. For any $b \ge a \ge 3$, we have

$$R_\mathrm{cyc}(P_a^\mathrm{mon}, P_b^\mathrm{mon}) = 1 + (a - 1)(b - 2).$$

The theorem preceding this conjecture supplies the matching lower bound, while the computational data suggest that the bound is sharp. The general equality remains open.