The cycle complement stable-set polynomial conjecture

Let $C_n$ be the cycle graph on $n$ vertices, let $p(C_n;z)$ denote its perfectly matchable set polynomial, and let $\overline{h}^*(C_n;z)$ denote the normalized $h^*$-polynomial of the stable set polytope of the complement of $C_n$. Cycle polynomial conjecture. For all $n\geq 4$,

$$\overline{h}^*(C_n; z) = p(C_n; z) + \frac{1+(-1)^{n+1}}{2}z^{(n+1)/2}.$$

The formula agrees with the explicitly computed even-cycle case and with the listed examples through $n=9$; the asserted general identity, particularly for odd cycles, remains open in the source.