The two-cyclic-factor conjecture for finite groups realized by 3-nets

Let $H$ be a finite group. A 3-net in $\mathbb{P}^2$ is a configuration of three line classes whose pairwise incidences define the group operation on $H$; one class is a pencil if all its lines pass through a common point.

Two-cyclic-factor conjecture. If $H$ can be realized by a 3-net in $\mathbb{P}^2$, then $H$ is a direct sum of at most two cyclic groups. If one class of the net is a pencil, then $H$ is cyclic.

The paper proves these conclusions under lower bounds on the orders of elements of $H$: at least $10$ for the first assertion and at least $7$ for the pencil assertion. The conjecture asks whether those order conditions can be removed; the realization theorem cited in the source establishes the converse direction for the relevant abelian groups.