The cohomology of $\text{Gr}_1(\mathbb{R}^{12,1})$

Poincaré polynomial: $$x^{11} y + x^{10} y + x^{9} y + x^{8} y + x^{7} y + x^{6} y + x^{5} y + x^{4} y + x^{3} y + x^{2} y + x y + 1$$ Generator count grid:
1 1 1 1 1 1 1 1 1 1 1 1
Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_1(\mathbb{R}^{12,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{7,1}\mathbb{M}_2\oplus\Sigma^{8,1}\mathbb{M}_2\oplus\Sigma^{9,1}\mathbb{M}_2\oplus\Sigma^{10,1}\mathbb{M}_2\oplus\Sigma^{11,1}\mathbb{M}_2.$$