Periodicity 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 $\boldsymbol\pi$ denote the distinguished central element used in the source. Periodicity conjecture. For every ${\bf w}\in B^+$, one has

$$[H_c^*({\bf X}(\boldsymbol\pi {\bf w}))_\rho]=h^{4N-2A_\rho}t^{2N-a_\rho-A_\rho}[H_c^*({\bf X}({\bf w}))_\rho].$$

This is one of several periodicity conjectures, verified in rank $2$ and for the elements whose cohomology is computed in the paper.