Cyclic Ramsey conjecture for monotone cycles and alternating paths

Let $C_a^\mathrm{mon}$ be a monotone cycle of order $a$ and $P_b^\mathrm{alt}$ an alternating path of order $b$. Cyclic monotone-cycle–alternating-path conjecture. For any $a \ge 2$ and $b \in \mathbb{N}$, we have

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

The claim is proposed after computational experiments, alongside the ordered version above. The general cyclic equality is not proved and remains open.