Directed Sidorenko conjecture for oriented graphs

Let $B$ be an oriented graph, let $G$ be an oriented graph, and let $\vec K_2$ denote the oriented complete graph on two vertices. The directed Sidorenko property means

$$t(B,G)\geq t(\vec K_2,G)^{e(B)}.$$

Directed Sidorenko conjecture. If $B$ is a bipartite oriented graph with a homomorphism $B\to\vec K_2$, then $B$ has the directed Sidorenko property. This is a directed analogue of Sidorenko's conjecture. The stated homomorphism condition is necessary for a bipartite oriented graph to have the directed Sidorenko property, and the paper conjectures that it is sufficient; the conjecture is related to the asymmetric undirected Sidorenko conjecture.