The discreteness conjecture for monodromy of reduced oriented networks

Let $N$ be an oriented network, and call it reduced when it satisfies the paper's reducedness condition. Let the monodromy action on $N$ be the action induced by the network's monodromy transformations. Discreteness conjecture. When an oriented network $N$ is reduced, the monodromy action on $N$ is the action of a discrete group. The paper indicates that Theorem 4.4 provides evidence for this conjecture, but the general assertion remains unresolved in the supplied text.