Kazhdan-Lusztig left cells in type $B_n$ for intermediate parameters

Using a characterization of a generalized $\tau$-invariant for intermediate parameter Hecke algebras in type $B_n$, we verify a conjectural description of Kazhdan-Lusztig cells in this setting due to C. Bonnaf'e, L. Iancu, M. Geck, and T. Lam.