Hahn–Hell–Poljak conjecture for odd wheels

Let $W_t$ be the join of the cycle $C_t$ with $K_1$, and let $d4b0(G)$ denote the ultimate independence ratio of a graph $G$. For odd wheels, $d4b0(W_{2t+1})$ is bounded below by $1/4$ because $d4b0(G)\geq 1/\chi(G)$ and $\chi(W_{2t+1})=4$. Hahn–Hell–Poljak's conjecture. For all integers $t\geq 1$,

$$\mathscr{I}(W_{2t+1})=\frac{1}{4}.$$

Determining the ultimate independence ratio of odd wheels is a central open problem in this setting; the $5$-wheel is identified as the smallest graph whose ratio is unknown.