The AutoKron can't determine the cohomology of $\text{Gr}_4(\mathbb{R}^{10,1})$.
There are 16 possibilities.
Here are their Poincaré polynomials:
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + x^{16} y^{4} + 11 x^{16} y^{3} + x^{16} y^{2} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + x^{9} y^{3} + 12 x^{9} y^{2} + x^{9} y + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + x^{16} y^{4} + 12 x^{16} y^{3} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + x^{9} y^{3} + 12 x^{9} y^{2} + x^{9} y + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + 12 x^{16} y^{3} + x^{16} y^{2} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + x^{9} y^{3} + 12 x^{9} y^{2} + x^{9} y + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + 13 x^{16} y^{3} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + x^{9} y^{3} + 12 x^{9} y^{2} + x^{9} y + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + x^{16} y^{4} + 11 x^{16} y^{3} + x^{16} y^{2} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + x^{9} y^{3} + 13 x^{9} y^{2} + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + x^{16} y^{4} + 12 x^{16} y^{3} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + x^{9} y^{3} + 13 x^{9} y^{2} + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + 12 x^{16} y^{3} + x^{16} y^{2} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + x^{9} y^{3} + 13 x^{9} y^{2} + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + 13 x^{16} y^{3} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + x^{9} y^{3} + 13 x^{9} y^{2} + 11 x^{8} y^{2} + 2 x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + x^{16} y^{4} + 11 x^{16} y^{3} + x^{16} y^{2} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + 13 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + x^{16} y^{4} + 12 x^{16} y^{3} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + 13 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + 12 x^{16} y^{3} + x^{16} y^{2} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + 13 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + 13 x^{16} y^{3} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + 2 x^{10} y^{3} + 14 x^{10} y^{2} + 13 x^{9} y^{2} + x^{9} y + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + x^{16} y^{4} + 11 x^{16} y^{3} + x^{16} y^{2} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + 14 x^{9} y^{2} + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + x^{16} y^{4} + 12 x^{16} y^{3} + 12 x^{15} y^{3} + 2 x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + 14 x^{9} y^{2} + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + x^{17} y^{4} + 9 x^{17} y^{3} + 12 x^{16} y^{3} + x^{16} y^{2} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + 14 x^{9} y^{2} + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
$$x^{24} y^{4} + x^{23} y^{4} + 2 x^{22} y^{4} + 2 x^{21} y^{4} + x^{21} y^{3} + 3 x^{20} y^{4} + 2 x^{20} y^{3} + 2 x^{19} y^{4} + 4 x^{19} y^{3} + 2 x^{18} y^{4} + 7 x^{18} y^{3} + 10 x^{17} y^{3} + 13 x^{16} y^{3} + 13 x^{15} y^{3} + x^{15} y^{2} + 11 x^{14} y^{3} + 5 x^{14} y^{2} + 9 x^{13} y^{3} + 7 x^{13} y^{2} + 7 x^{12} y^{3} + 11 x^{12} y^{2} + 4 x^{11} y^{3} + 12 x^{11} y^{2} + x^{10} y^{3} + 15 x^{10} y^{2} + 14 x^{9} y^{2} + 12 x^{8} y^{2} + x^{8} y + 7 x^{7} y^{2} + 3 x^{7} y + 5 x^{6} y^{2} + 4 x^{6} y + 2 x^{5} y^{2} + 4 x^{5} y + x^{4} y^{2} + 4 x^{4} y + 3 x^{3} y + 2 x^{2} y + x y + 1$$
Here are the corresponding generator grids to these 16 possibilities: