The Berge graph decomposition conjecture

Let $G$ be a finite simple graph. A Berge graph is a graph containing neither an odd hole nor the complement of an odd hole. A graph is basic if it belongs to one of the four classes consisting of bipartite graphs, line graphs of bipartite graphs, or their complements. A skew partition is a partition of the vertices into four nonempty sets $A,B,C,D$ with all possible edges between $A$ and $B$ and no edges between $C$ and $D$. A homogeneous pair is a partition of $V(G)$ into $A_1,A_2,B$ with $|A_1|+|A_2|\geq 3$, $|B|\geq 2$, and every vertex of $B$ adjacent to a vertex of $A_i$ adjacent to all vertices of $A_i$, for $i=1,2$. A 2-join is the decomposition notion used in the source. Decomposition conjecture. Every Berge graph $G$ is basic or has a skew partition or a homogeneous pair, or $G$ or $\bar G$ has a 2-join.

The source attributes this structural conjecture to Conforti, Cornuéjols, Robertson, Seymour, Thomas and Vušković (2001), and says that it implies the Strong Perfect Graph Conjecture. The supplied text gives no resolution status.