Conjecture A on Kazhdan–Lusztig cells and domino insertion

Let $W_n$ be the Coxeter group of type $B_n$ with weight function $L_{a,b}$, and fix $r\geqslant 0$. Let $P^r(w)$ and $Q^r(w)$ be the insertion and recording standard domino tableaux associated to $w\in W_n$ by domino insertion with respect to the $2$-core $\delta_r$.

Conjecture A. Assume that $\Gamma=\mathbb{Z}$, $a=2$ and $b=2r+1$. Then, for $w,w'\in W_n$, they lie in the same Kazhdan–Lusztig left cell if and only if $Q^r(w)=Q^r(w')$, in the same right cell if and only if $P^r(w)=P^r(w')$, and in the same two-sided cell if and only if $P^r(w)$, $Q^r(w)$, $P^r(w')$, and $Q^r(w')$ all have the same shape.

These conjectures give an explicit combinatorial description of the Kazhdan–Lusztig cells in this integral parameter case. They had been verified by explicit computation for $n\leqslant 6$; the paper presents relative results supporting them.