Algebraic-cycle conjecture for Springer fibres

Let $X=T^*(G/B)$ be the Springer resolution of the nilpotent cone $Y={\mathcal N}\subset{\mathfrak g}^*$ in the coadjoint representation of a semisimple algebraic group $G$, let $y\in Y$, and let $E_y=\pi^{-1}(y)$ be the corresponding Springer fibre. For each integer $k$, consider its singular cohomology group $H^k(E_y,\mathbb Z)$. Algebraic-cycle conjecture for Springer fibres. In the assumptions of the symplectic-resolution theorem, $H^k(E_y,\mathbb Z)$ should be trivial for odd $k$, and for even $k$ it should be spanned by cohomology classes of algebraic cycles. This strengthens the stated Hodge-theoretic conclusion for Springer fibres: the source notes that the assertion is known for the Springer resolution by work of C. de Concini, G. Lusztig and C. Procesi, but presents it as a proposed statement in the general assumptions of the theorem.