Karpenko's conjecture on Chow rings and topological filtrations

Let $G$ be a split semisimple linear algebraic group over a field $F$, let $B\subset G$ be a Borel subgroup, and let $E$ be a generic $G$-torsor. Write $X=E/B$ and let

$$\varphi_X\colon\mathrm{CH}^*(X)\twoheadrightarrow \mathrm{gr}_{\tau}\mathrm{K}_0(X)$$

be the canonical morphism from the Chow ring to the associated graded ring of the topological filtration. Karpenko's conjecture. The morphism $\varphi_X$ is an isomorphism. This conjecture asks whether the canonical approximation of the Chow ring by the topological filtration on the Grothendieck ring is exact for generically twisted varieties of complete flags. It has been disproved for some spinor groups, including new counterexamples presented in this paper, so the conjecture is refuted.