The cohomology of $\text{Gr}_2(\mathbb{R}^{10,5})$

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