Mixed cyclic Ramsey conjecture for alternating and monotone paths

Let $P_a^\mathrm{alt}$ be an alternating path of order $a$ and $P_b^\mathrm{mon}$ a monotone path of order $b$. Mixed path conjecture. For any $a \ge 3$ and $b \ge 4$, we have

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

The claim is motivated by the computed mixed cyclic Ramsey numbers and by the discussion that the case $b=3$ is rotationally equivalent to the alternating-path case. The asserted formula remains open.