Classification conjecture for Griffiths semi-positive quasi-Kähler flag manifolds

Let $M$ be a flag manifold with an almost-complex structure $J$ and a quasi-Kähler metric $\lambda$ whose curvature is Griffiths semi-positive. Let $(M_i,g_i)$ denote Hermitian symmetric spaces with their unique invariant metrics. Classification conjecture. Then $(M,J,\lambda)$ is isometrically biholomorphic to

$$(M_1,g_1)\times\cdots\times(M_p,g_p),$$

where each $M_i$ is a Hermitian symmetric space and $g_i$ is its unique invariant metric. The paper establishes nonexistence in several families, including classical simple Lie groups, maximal flag manifolds, and $G_2$; the proposed classification in the remaining cases is open.