Torus case of the Deligne–Lusztig restriction conjecture

Let $W$ be the Weyl group, let $w\in W$, let $\dot w$ be a representative of $w$, and let $\psi:U_0\to\mathcal{O}^{\times}$ be a regular linear character. Torus restriction conjecture. There is an isomorphism

$${}^*\mathcal{R}_{\dot w}\Gamma_{\psi,G}\simeq\mathcal{O}{\mathbf{T}}_0^{wF}[-l(w)].$$

This is the torus specialization of the preceding conjecture and is known in the Coxeter cases treated in the paper, but is not established in general.