The virtual cohomology decomposition conjecture for the full twist

Let $W$ be the Weyl group, let $F$ be the Frobenius automorphism, and let ${\boldsymbol\pi}$ be the positive generator of the center of the pure braid group. Let ${\bf X}({\boldsymbol\pi})$ be the associated Deligne–Lusztig variety, let $\operatorname{Irr}(W^F)$ denote the irreducible characters of $W^F$, let $\rho_\chi$ be the corresponding characters of ${{\bf G}^F}$, and let $\sigma$ be the semilinear automorphism sending $x^{1/2}$ to $-x^{1/2}$, with $\sigma(\chi)_q$ denoting the induced specialized Hecke character. Virtual cohomology decomposition conjecture.

$$\sum_i(-1)^i H^i_c({\bf X}({\boldsymbol\pi}))= \sum_{\chi\in\operatorname{Irr}(W^F)}\rho_\chi\otimes \sigma(\chi)_q.$$

The source states that this identity would imply the preceding character and specialness conjecture for ${\bf y}={\boldsymbol\pi}$; no resolution is supplied for the displayed assertion.