Duality 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 $\overline{P}$ be the involution of $\mathbb Z[t^{\pm1/2},h^{\pm1}]$ defined by

$$h\mapsto h^{-1},\qquad t^{1/2}\mapsto t^{-1/2}.$$

Duality periodicity conjecture. If ${\bf w}\in B^+$ satisfies ${\bf w}\preccurlyeq\boldsymbol\pi^n$, then

$$\langle H_c^*({\bf X}({\bf w}^{-1}\boldsymbol\pi^n)),\rho\rangle=(h^{4N-2A_\rho}t^{2N-a_\rho-A_\rho})^n\overline{\langle H_c^*({\bf X}({\bf w})),\rho\rangle}.$$

This is a companion periodicity assertion to the preceding conjecture and is supported by the rank-$2$ calculations described in the source.