Rédei's additive decomposition conjecture in three-dimensional prime vector spaces

Let $A,B\subseteq\mathbb{F}_p^3$ be subsets with $0\in A$ and $0\in B$. The notation $A+B$ denotes the sumset $\{a+b:a\in A,\ b\in B\}$, and $\langle A\rangle$ denotes the linear span of $A$. Rédei's conjecture. If $A+B=\mathbb{F}_p^3$, then $\langle A\rangle\ne\mathbb{F}_p^3$ or $\langle B\rangle\ne\mathbb{F}_p^3$. The source describes this as a renowned conjecture and explains implications from the cylinder conjecture toward it, so its resolution is not supplied here.