  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.
