Module structure of cells in unequal parameter Hecke algebras

A conjecture of C. Bonnaf'e, M. Geck, L. Iancu, and T. Lam parameterizes Kazhdan-Lusztig left cells for unequal parameter Hecke algebras in type $B_n$ by families of standard domino tableaux of arbitrary rank. Relying on a family of properties outlined by G. Lusztig and the recent work of C. Bonnaf'e, we verify the conjecture and describe the structure of each cell as a module for the underlying Weyl group.