Decomposability conjecture for conjugation-free line arrangements and groups

Let $\mathcal{L}$ be a line arrangement, and let $G$ be a group. They are conjugation-free when their defining relations have no conjugations; they are decomposable when their associated graph is cycle-separated. Decomposability conjecture. If $\mathcal{L}$, respectively $G$, is a conjugation-free line arrangement, respectively group, then $\mathcal{L}$, respectively $G$, is decomposable. The conjecture is motivated by computations for arrangements whose graphs are not cycle-separated; its general validity remains open.