The Hodge filtration compatibility conjecture for the affine Hecke equivalence

Let ${\mathcal{MH}}_{U\check{\ }}({ G\check{\ }}/{B\check{\ }})$ be the category of mixed Hodge modules on ${ G\check{\ }}/{B\check{\ }}$ equivariant with respect to ${U\check{\ }}$. For $\tilde M$ in this category, let $M$ be its underlying $D$-module and let $gr(\tilde M)$ be the associated graded object of its canonical good filtration. Then the Hodge filtration compatibility conjecture. There is a canonical isomorphism

$$gr(\tilde M)\otimes {\mathcal O}(-\rho)\cong\Psi_{I^0 I}(\Upsilon(M)).$$

This conjecture relates mixed Hodge module filtrations to the geometric realization of the affine Hecke algebra and can be compared with results of Ben-Zvi and Nadler.