The cohomology of $\text{Gr}_3(\mathbb{R}^{7,1})$

Poincaré polynomial: $$x^{12} y^{3} + x^{11} y^{3} + x^{10} y^{3} + x^{10} y^{2} + x^{9} y^{3} + 2 x^{9} y^{2} + 4 x^{8} y^{2} + 4 x^{7} y^{2} + 4 x^{6} y^{2} + x^{6} y + 2 x^{5} y^{2} + 2 x^{5} y + x^{4} y^{2} + 3 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$ Generator count grid:
1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 4 4 4 1 2 2 2 2 1 3 3 3 3 3 3 2 2 1 1
Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_3(\mathbb{R}^{7,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2\oplus\Sigma^{8,2}\mathbb{M}_2\oplus\Sigma^{9,2}\mathbb{M}_2\oplus\Sigma^{9,3}\mathbb{M}_2\oplus\Sigma^{10,2}\mathbb{M}_2\oplus\Sigma^{10,3}\mathbb{M}_2\oplus\Sigma^{11,3}\mathbb{M}_2\oplus\Sigma^{12,3}\mathbb{M}_2.$$