Directed forcing conjecture for oriented graphs

Let $B$ be an oriented graph, let $\overline{B}$ be its underlying undirected graph obtained by forgetting edge directions, and call $B$ directed-forcing when it has the directed forcing property. Directed forcing conjecture. If $B$ has a homomorphism to an edge and $\overline{B}$ contains a cycle, then $B$ has the directed forcing property. The paper proves that a homomorphism to an edge and a cycle in the underlying graph are necessary conditions for directed forcing. It conjectures that these two conditions are also sufficient, leaving the characterization open.