Verdier duality corresponds to the canonical-basis involution

Let ${\mathcal{P}}$ be the category of semisimple perverse sheaves on ${\mathfrak{T}(\mathbf{d})}'$, let $\mathcal{D}:{\mathcal{P}}\to{\mathcal{P}}$ be Verdier duality, and let $\theta:{\mathcal{P}}\to\mathcal{T}(\mathbf{d})$ be the map defined by

$$(\theta(\mathbf{B}^{\bullet}))(x)=\sum_i(-1)^i q^i(\mathbf{B}_x^i).$$

Let $\Psi^{(k)}$ denote the involution on $\mathcal{T}(\mathbf{d})$ induced from the canonical-basis involution. Verdier-duality compatibility conjecture.

$$\theta\mathcal{D}=\Psi^{(k)}\theta.$$

This compatibility is proposed as the central idea for proving the intersection-sheaf characterization of the canonical basis; the supplied text does not state that it has been proved.