Parabolic convolution action conjecture

Let $\bmu$ be a parabolic character, and let $\sHg^c$ and $\sWg^c$ be the associated twisted equivariant perverse-sheaf categories. Convolution is the product operation on these categories. Parabolic convolution action conjecture. The category $\sHg^c$ acts on $\sWg^c$ by convolution. This is the categorical counterpart of the action of the corresponding Hecke algebra on the principal-series representation. The source says that the assertion specializes to a theorem in the unramified and regular cases; no general proof is supplied.