Woldar's purely singular splitting conjecture

Let $k$ be a positive integer and write $S(k)=\{1,2,\dots,k\}$. Let $G$ be a finite abelian group. Say that $S(k)$ splits $G$ if there is a subset $T\subseteq G$ such that the translates represented by $S(k)\cdot T$ give $G\setminus\{0\}$, and say that the splitting is purely singular if every prime divisor of $|G|$ divides some element of $S(k)$. Woldar's conjecture. If $S(k)$ splits a finite abelian group $G$ purely singularly, then $G$ is one of $\mathbb{Z}_1$, $\mathbb{Z}_{k+1}$, or $\mathbb{Z}_{2k+1}$. Woldar proposed this classification in 1995, and it was verified by Hickerson for all $k\leq 3000$; the conjecture concerns the remaining values of $k$.