Dartyge–Sárközy conjecture on additive decompositions of primitive elements

Let $p$ be a prime, let $\mathbb{F}_p$ be the finite field with $p$ elements, and let $P_p$ denote the set of primitive elements of $\mathbb{F}_p$. For subsets $A,B\subseteq\mathbb{F}_p$, write

$$A+B=\{a+b:a\in A,\ b\in B\}.$$

Dartyge–Sárközy conjecture. If $p$ is large enough, then

$$A+B\neq P_p$$

for any $A,B\subseteq\mathbb{F}_p$ with $|A|,|B|\ge 2$.

This conjecture concerns whether the set of primitive elements of a finite field can admit a nontrivial additive decomposition. The source attributes it to Dartyge and Sárközy; no resolution is supplied here.