The 2-torsion conjecture for real flag manifolds and R-spaces

Let $\operatorname{Fl}_{\mathcal{D}}$ be a real flag manifold, and let $H^*(\operatorname{Fl}_{\mathcal{D}};\mathbb{Z})$ denote its integral cohomology. More generally, let an $R$-space be a homogeneous space of the type considered in the paper. The 2-torsion conjecture. For any real flag manifold, and more generally any $R$-space, all torsion in

$$H^*(\operatorname{Fl}_{\mathcal{D}};\mathbb{Z})$$

has order $2$. This extends the theorem proved for even real flag manifolds and the classical result for real Grassmannians; the assertion for arbitrary real flag manifolds and $R$-spaces remains conjectural.