Classification conjecture for (1,2)-symplectic Borel metrics on full flag manifolds

Let $F(n)$ be a full flag manifold with $n\geq 5$, equipped with a complex structure $J$ and its associated tournament ${\cal T}(J)$. Let $ds^2_{\Lambda}$ denote the Borel metric. A 4-subtournament is a subtournament induced by four vertices of ${\cal T}(J)$; it is called transitive or irreducible according to its tournament type. Classification conjecture. The Borel metric $ds^2_{\Lambda}$ is (1,2)-symplectic if and only if every 4-subtournament of the associated tournament ${\cal T}(J)$ is transitive (class (4) in Figure 1) or irreducible (class (7) in Figure 1). The conjecture would classify the Borel metrics that are (1,2)-symplectic in terms of the associated tournament, extending the partial classification obtained in the preceding results; its resolution is not given in the supplied text.