Cyclic Ramsey conjecture for stars and monotone paths

Let $S_a$ be the start-central star with parameter $a$, and let $P_b^\mathrm{mon}$ be a monotone path of order $b$. Star–monotone-path conjecture. For any $a \ge 3$ and $b \ge 4$, we have

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

The paper presents this as a conjecture suggested by computational results; the lower-bound pattern is consistent with the path results, but the general equality remains open.