On generic flag varieties of Spin(11) and Spin(12)

Let X be the variety of Borel subgroups of a split semisimple algebraic group G over a field, twisted by a generic G-torsor. Conjecturally, the canonical epimorphism of the Chow ring $$\mathop {\mathrm {CH}}\nolimits X$$ onto the associated graded ring GK(X) of the topological filtration on the Grothendieck ring K(X) is an isomorphism. We prove the new cases $$G={\text {Spin}}(11)$$ and $$G={\text {Spin}}(12)$$ of this conjecture. On an equivalent note, we compute the Chow ring $$\mathop {\mathrm {CH}}\nolimits Y$$ of the highest orthogonal grassmannian Y for the generic 11- and 12-dimensional quadratic forms of trivial discriminant and Clifford invariant. In particular, we describe the torsion subgroup of the Chow group $$\mathop {\mathrm {CH}}\nolimits Y$$ and determine its order which is equal to $$16\;777\; 216$$ . On the other hand, we show that the Chow group $$\mathop {\mathrm {CH}}\nolimits _0Y$$ of 0-cycles on Y is torsion-free.