Goresky–Kottwitz–Macpherson purity conjecture for affine Springer fibers

Let $k=\mathbf{F}_{q}$, let $G$ be a connected reductive group split over $k$, and let $\gamma\in \mathfrak{t}(\mathcal{O})$ be regular, with affine Springer fiber

$$\mathscr{X}_{\gamma}=\{g\in G(F)/K\mid\operatorname{Ad}(g^{-1})\gamma\in\mathfrak{g}(\mathcal{O})\}.$$

Here $F=k((\epsilon))$, $\mathcal{O}=k[[\epsilon]]$, $K=G(\mathcal{O})$, and $\mathscr{X}_{\gamma,\bar{k}}$ denotes base change to an algebraic closure. Goresky–Kottwitz–Macpherson purity conjecture. The cohomology of $\mathscr{X}_{\gamma}$ is pure in the sense of Deligne: the eigenvalues of Frobenius $\operatorname{Fr}_{q}$ on $H^{i}(\mathscr{X}_{\gamma,\bar{k}},\overline{\mathbf{Q}}_{l})$ have absolute value $q^{i/2}$ under every embedding $\overline{\mathbf{Q}}_{l}\to\mathbf{C}$. This purity conjecture is known in some cases, notably when affine pavings are available, but is not resolved in general.