Cyclic Ramsey conjecture for complete graphs and alternating paths

Let $K_a$ be a complete graph of order $a$ and $P_b^\mathrm{alt}$ an alternating path of order $b$. Complete-graph cyclic conjecture. For any $a, b \in \mathbb{N}$, we have

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

The claim is suggested by the computational data and contrasts with the proposed larger ordered Ramsey numbers. The cyclic equality remains open.