The cohomology of $\text{Gr}_2(\mathbb{R}^{13,1})$

Poincaré polynomial: $$x^{22} y^{2} + x^{21} y^{2} + 2 x^{20} y^{2} + 2 x^{19} y^{2} + 3 x^{18} y^{2} + 3 x^{17} y^{2} + 4 x^{16} y^{2} + 4 x^{15} y^{2} + 5 x^{14} y^{2} + 5 x^{13} y^{2} + 5 x^{12} y^{2} + x^{12} y + 4 x^{11} y^{2} + 2 x^{11} y + 4 x^{10} y^{2} + 2 x^{10} y + 3 x^{9} y^{2} + 2 x^{9} y + 3 x^{8} y^{2} + 2 x^{8} y + 2 x^{7} y^{2} + 2 x^{7} y + 2 x^{6} y^{2} + 2 x^{6} y + x^{5} y^{2} + 2 x^{5} y + x^{4} y^{2} + 2 x^{4} y + 2 x^{3} y + 2 x^{2} y + x y + 1$$ Generator count grid:
1 1 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 1 4 4 4 4 2 2 4 4 4 4 2 2 3 3 3 2 2 3 3 3 2 2 2 2 2 2 2 2 2 2 1 2 2 1 2 2 2 2 2 2 1 1
Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_2(\mathbb{R}^{13,1}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{3,1}\mathbb{M}_2\oplus\Sigma^{4,1}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{5,1}\mathbb{M}_2\oplus\Sigma^{5,2}\mathbb{M}_2\oplus\Sigma^{6,1}\mathbb{M}_2\oplus\Sigma^{6,2}\mathbb{M}_2\oplus\Sigma^{7,1}\mathbb{M}_2\oplus\Sigma^{7,2}\mathbb{M}_2\oplus\Sigma^{8,1}\mathbb{M}_2\oplus\Sigma^{8,2}\mathbb{M}_2\oplus\Sigma^{9,1}\mathbb{M}_2\oplus\Sigma^{9,2}\mathbb{M}_2\oplus\Sigma^{10,1}\mathbb{M}_2\oplus\Sigma^{10,2}\mathbb{M}_2\oplus\Sigma^{11,1}\mathbb{M}_2\oplus\Sigma^{11,2}\mathbb{M}_2\oplus\Sigma^{12,1}\mathbb{M}_2\oplus\Sigma^{12,2}\mathbb{M}_2\oplus\Sigma^{13,2}\mathbb{M}_2\oplus\Sigma^{14,2}\mathbb{M}_2\oplus\Sigma^{15,2}\mathbb{M}_2\oplus\Sigma^{16,2}\mathbb{M}_2\oplus\Sigma^{17,2}\mathbb{M}_2\oplus\Sigma^{18,2}\mathbb{M}_2\oplus\Sigma^{19,2}\mathbb{M}_2\oplus\Sigma^{20,2}\mathbb{M}_2\oplus\Sigma^{21,2}\mathbb{M}_2\oplus\Sigma^{22,2}\mathbb{M}_2.$$