Non-sharpness of the Paley graph Shannon-capacity bounds

Let $k\geq3$ and let $p$ be a prime satisfying $p\equiv1\pmod{k}$. Write $\Theta(G)$ for the Shannon capacity of a graph $G$, and let $\operatorname{Paley}_{k}(\mathbb{F}_{p})$ be the generalized Paley graph over $\mathbb{F}_p$.

Paley Shannon-capacity conjecture. There exist constants $a_k,b_k>0$ such that, for every prime $p\equiv1\pmod{k}$,

$$p^{\frac{1}{2}+a_k}\leq\Theta\left(\operatorname{Paley}_{k}(\mathbb{F}_{p})\right)\leq p^{1-\frac{1}{k}-b_k}.$$

The conjecture asserts that neither currently known bound is sharp for prime fields when $k\geq3$, leaving a positive power gap from both sides.