Bonnafé–Geck–Iancu–Lam conjecture on left cells in type B_n

Let $W_n$ be a Weyl group of type $B_n$ with weight function $L$, and let $s=b/a$ be its parameter. For each non-negative integer $r$, let $G_r:W_n\to SDT_r(n)\times SDT_r(n)$ be the generalized Robinson–Schensted map, and write $G_r(x)=(S_r(x),T_r(x))$. The irreducible combinatorial left cells of rank $r$ are the equivalence classes defined by $T_r(x)=T_r(y)$; the reducible combinatorial left cells of rank $r$ are the equivalence classes defined by $T_r(y)=MT(T_r(x),C)$ for some set $C$ of non-core open cycles.

Bonnafé–Geck–Iancu–Lam conjecture. If $s\notin\mathbb{N}$, the Kazhdan–Lusztig left cells coincide with the irreducible combinatorial left cells of rank $\lfloor s\rfloor$. If $s\in\mathbb{N}$, the Kazhdan–Lusztig left cells coincide with the reducible combinatorial left cells of rank $s-1$.

This conjecture describes the Kazhdan–Lusztig left-cell decomposition in type $B_n$ using generalized Robinson–Schensted algorithms and standard domino tableaux. The paper states that it verifies the conjecture, so the claim is recorded as solved.