Horton's strong starter conjecture for finite abelian groups

Let $G$ be an abelian group of odd order $g\geq 3$. A strong starter in $G$ is a starter whose pair sums are distinct and nonzero. Horton's conjecture. There is a strong starter in $G$ if and only if $G\neq \mathbb{Z}_3,\mathbb{Z}_5,\mathbb{Z}_9$ or $\mathbb{Z}_3\times\mathbb{Z}_3$. This generalizes the cyclic special case and is described in the source as still far from solved.