Ennola duality conjecture for Deligne–Lusztig cohomology

Let $N=l(w_0)$, let $\rho\in{\mathcal E}({\bf G}^F,1)$, and let $A_\rho$ and $a_\rho$ be respectively the degree and valuation of the polynomial in $q$ giving $\rho(1)$. Let $E$ be the Ennola involution of ${\mathcal E}({\bf G}^F,1)$. Ennola duality conjecture. Assume that $w_0$ is central in $W$. For every ${\bf w}\in B^+$, one has

$$\langle H_c^*({\bf X}({\bf w}_0{\bf w})),\rho\rangle=h^{2N-A_\rho}t^{N-(a_\rho+A_\rho)/2}\langle H_c^*({\bf X}({\bf w})),E(\rho)\rangle.$$

The conjecture relates cohomology after multiplication by the longest Weyl-group element to the Ennola involution; the surrounding periodicity claims are verified in rank $2$ and in the cases computed in the paper.