Are there graphs where $\alpha(G) = \chi^*(\bar{G}) < \chi(\bar{G})$???

Here, $\chi^*(\bar{G})$ is the fractional chromatic number, which I believe is also equal to the fractional independence number by the duality of linear programming. The point is, since for all graphs we have

$$\alpha(G) \leq \Theta(G) \leq \chi^*(\bar{G}) \leq \chi(\bar{G})$$

where $\Theta(G)$ is the Shannon capacity, I'm wondering if there are graphs where the Shannon capacity is determined by $\alpha(G)$ and $\chi^*(\bar{G})$, but not by $\alpha(G)$ and $\chi(\bar{G})$.

Thanks