Purity conjecture for the fundamental domain of an affine Springer fiber

Let $\mathscr{X}_{\gamma}$ be an affine Springer fiber, and let $F_{\gamma}$ be a projective algebraic fundamental domain of finite type for the action of the relevant free abelian group on $\mathscr{X}_{\gamma}$. Its cohomology is pure in the sense of Deligne if it has the expected Frobenius weights.

Fundamental-domain purity conjecture. The cohomology of $F_{\gamma}$ is pure in the sense of Deligne.

The paper reduces the affine-Springer-fiber purity conjecture to this assertion and proposes to prove it by constructing affine pavings of $F_{\gamma}$. Its general status is open.