Factoring elementary groups of prime cube order into subsets

Let $p$ be a prime and let $G$ be the $3$-fold direct product of the cyclic group of order $p$. Rédei conjectured if $G$ is the direct product of subsets $A$ and $B$, each of which contains the identity element of $G$, then either $A$ or $B$ does not generate all of $G$. The paper verifies Rédei's conjecture for $p\leq 11$.