Conjecture on t-exactness of the long intertwining transform

Let $G$ be the reductive group, with $U$ and $U^-$ opposite maximal unipotent subgroups and $T$ a maximal torus, and let

$$\textnormal{HC}_*: D(G \backslash G) \to D(G \backslash (G/U \times G/U) / T)$$

be the Harish-Chandra transform and

$$R_!: D(G \backslash (G/U \times G/U) / T) \to D(G \backslash (G/U \times G/U^-) / T)$$

be the long intertwining transform. The t-exactness conjecture. The functor $R_! \circ \textnormal{HC}_*$ is $t$-exact. This conjecturally generalizes the known $t$-exactness of the composition on character sheaves to the relevant categories of equivariant sheaves.