Conjecture on Kazhdan–Lusztig cells at integral parameter ratios

Let $W_n$ be the Coxeter group of type $B_n$, and for each $r\geqslant 0$ let the left, right, and two-sided $r$-cells be the fibers determined by the domino-insertion tableaux $Q^r(w)$, $P^r(w)$, and their common shape, respectively.

Integral-ratio cell conjecture. Assume that $b=ra$ for some $r\geqslant 1$. Then the Kazhdan–Lusztig left, right, and two-sided cells of $W_n$ are the smallest subsets of $W_n$ that are simultaneously unions of the corresponding $(r-1)$-cells and $r$-cells.

This addresses the parameter values excluded from the open intervals in Conjectures A and A+. It proposes that the cells at an integral ratio are obtained by combining the adjacent domino-cell decompositions; the paper leaves this assertion conjectural.