Berge's conjecture on alpha-diperfect digraphs

Let $D$ be a digraph. A stable set is a set of pairwise non-adjacent vertices, and a path partition is a collection of disjoint paths containing every vertex of $D$ exactly once. A stable set and a path partition are orthogonal when each path contains exactly one vertex of the stable set. A digraph satisfies the $lpha$-property when every maximum stable set has an orthogonal path partition, and it is $lpha$-diperfect when every induced subdigraph satisfies the $lpha$-property. An anti-directed odd cycle is a non-oriented cycle $C=x_1x_2\dots x_{2k+1}x_1$, where $k\geq 2$, and each of $x_1,x_2,x_3,x_4,x_6,x_8,\ldots,x_{2k}$ is either a source or a sink. Berge's conjecture. A digraph $D$ is $lpha$-diperfect if and only if $D$ does not contain an anti-directed odd cycle as an induced subdigraph. Berge's obstruction gives a forbidden-subdigraph direction for alpha-diperfectness; the supplied text does not report a resolution of the characterization.