The AutoKron can't determine the cohomology of $\text{Gr}_3(\mathbb{R}^{7,3})$.
There are 14 possibilities.
Here are their Poincaré polynomials:
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + x^{5} y^{5} + 4 x^{6} y^{3} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + x^{5} y^{5} + 3 x^{6} y^{3} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{4} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{4} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + 4 x^{5} y^{3} + x^{4} y^{4} + x^{4} y^{3} + 2 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + 4 x^{5} y^{3} + x^{4} y^{4} + x^{4} y^{3} + 2 x^{4} y^{2} + 3 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + x^{3} y^{3} + 2 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + x^{3} y^{3} + 2 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + 4 x^{5} y^{3} + 2 x^{4} y^{3} + 2 x^{4} y^{2} + x^{3} y^{3} + 2 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + 4 x^{5} y^{3} + 2 x^{4} y^{3} + 2 x^{4} y^{2} + x^{3} y^{3} + 2 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + 2 x^{2} y^{2} + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + x^{5} y^{4} + 3 x^{5} y^{3} + x^{4} y^{3} + 3 x^{4} y^{2} + 3 x^{3} y^{2} + 2 x^{2} y^{2} + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + x^{7} y^{5} + 3 x^{7} y^{4} + x^{6} y^{4} + 4 x^{6} y^{3} + 4 x^{5} y^{3} + 2 x^{4} y^{3} + 2 x^{4} y^{2} + 3 x^{3} y^{2} + 2 x^{2} y^{2} + x y + 1$$
$$x^{12} y^{6} + x^{11} y^{6} + 2 x^{10} y^{5} + 3 x^{9} y^{5} + x^{8} y^{5} + 3 x^{8} y^{4} + 4 x^{7} y^{4} + 2 x^{6} y^{4} + 3 x^{6} y^{3} + 4 x^{5} y^{3} + 2 x^{4} y^{3} + 2 x^{4} y^{2} + 3 x^{3} y^{2} + 2 x^{2} y^{2} + x y + 1$$
Here are the corresponding generator grids to these 14 possibilities: