Integrability conjecture for invariant generalized complex structures on maximal real flag manifolds

Let a maximal real flag manifold be a homogeneous space associated with a real simple Lie group of type $C_4$ or $D_4$, and let $K$ denote the acting compact group. A $K$-invariant generalized almost complex structure is an invariant generalized almost complex structure under this action.

Integrability conjecture. No maximal real flag manifold of type $C_4$ or $D_4$ admits integrable $K$-invariant generalized almost complex structures.

The conjecture concerns the cases excluded from the paper's computational treatment because their $M$-equivalence class root subspaces have dimension $4$, making the integrability calculations substantially more difficult. It is motivated by the results obtained in the paper and by Theorem 2 of the cited work, but its resolution is left for future work.