Characterization of BE-diperfect digraphs by blocking odd cycles

Let $D$ be a digraph. A digraph is BE-diperfect according to the definition in the source, and a blocking odd cycle is the obstruction defined there.

Proposed characterization. A digraph $D$ is BE-diperfect if, and only if, $D$ has no blocking odd cycle as an induced subdigraph.

The authors propose this characterization as analogous to Berge's proposed characterization of alpha-diperfect digraphs. The supplied text does not indicate whether it has been resolved.