Cyclic Ramsey conjecture for stars and alternating paths

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

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

The formula is suggested by the computed cyclic Ramsey numbers of stars versus alternating paths. Its validity beyond the computed range remains open.