Consider a split algebraic group $G$ of type $E_7$ over a field of characteristic zero. It is known that some subgroups $P_i$ of $G$, which are called parabolic, have the property that the object $G/P_i =:X_i$ is a projective, homogeneous variety over $k$. These parabolic subgroups correspond to subsets of nodes in the Dynkin diagram of $G$. We use Bourbaki enumeration. We write Ch$(X)$ for the Chow ring of $X$ modulo rational equivalence and with $\mathbb{F}_2$ coefficients.

**Question:** What is the structure of Ch$(X_3)$ ?

I can find references for $X_1$ and $X_7$, but for the Chow ring of $X_3$, which is probably way more complicated, i cant find any results. It would already be great to know how many generators there are and what their dimension is.