Path-graph incompatibility robustness conjecture

Let $P_n$ be the path graph on $n$ vertices, and let $C_n$ be the cycle graph on $n$ vertices. The incompatibility robustness of a graph is denoted by $\eta(\cdot)$.

Path-graph robustness conjecture. For all $n\geq 1$,

$$\eta(P_{2n})=\eta(P_{2n+1})=\eta(C_{2n+2})=\frac{2}{2n+2}\csc\Big(\frac{\pi}{2n+2}\Big).$$

The prediction is motivated by comparisons with exact values obtained by semidefinite programming and by the asymptotically tight bounds previously established for paths. It remains open whether these equalities hold for every $n\geq 1$.