Cyclic Ramsey conjecture for nested matchings

Let $M_a^\mathrm{nest}$ and $M_b^\mathrm{nest}$ denote nested matchings with parameters $a$ and $b$. Nested-matching conjecture. Let $a, b \ge 4$ be divisible by four. Then

$$R_\mathrm{cyc}(M_a^\mathrm{nest}, M_b^\mathrm{nest}) = a + b - 3.$$

A construction in the paper gives a lower bound of $a+b-4$ under the same divisibility conditions, while the computational results suggest the stronger value $a+b-3$. The equality remains open.