The sharp sum-product conjecture for dense subsets of finite fields

Let $p$ be prime and let $A\subset\mathbb{F}_p$ satisfy $|A|=o(p)$ and $|A|\geq p^{1-c}$ for some sufficiently small constant $c>0$. Sharp sum-product conjecture. Then

$$\max(|A+A|,|A\cdot A|)\geq (1+o(1))(2p|A|)^{1/2}.$$

This conjecture predicts the asymptotically sharp lower bound suggested by the preceding construction and the small-density behavior of the function $f(\alpha)$; it concerns sets whose sizes are close to, but still little-oh of, the field size. The supplied text does not indicate that the conjecture has been resolved.