Stratified cohomology conjecture for Deligne–Lusztig varieties

Let $W$ be the Weyl group, let $w\in W$, and let $Y_x(\dot w)$ be the stratum defined by

$$Y_x(\dot w)=\{gU_0\in B_0xU_0/U_0\mid g^{-1}F(g)\in U_0\dot wU_0\}.$$

For a $G$-regular linear character $\psi:U_0\to\mathcal{O}^{\times}$, let $\mathcal{O}_\psi$ be the associated $\mathcal{O}U_0$-module. Stratified cohomology conjecture. There are isomorphisms in $D^b(\mathcal{O}{\mathbf{T}}_0^{wF})$:

$$R\operatorname{Hom}\nolimits^\bullet_{\mathcal{O}U_0}(R\Gamma_c(Y_x(\dot w),\mathcal{O}),\mathcal{O}_\psi)\simeq \begin{cases} \mathcal{O}{\mathbf{T}}_0^{wF} & \text{if }x=w_\Delta,\\ 0 & \text{otherwise.} \end{cases}$$

This conjecture refines the torus case by specifying which strata contribute to the cohomology. The source proposes it as a refinement; no general proof or disproof is supplied.