Geck–Lam conjecture on Kazhdan–Lusztig cells and r-cycle equivalence

Let $a,b$ be the parameters defining the relevant Kazhdan–Lusztig cells, let $r\geq 1$, and let signed permutations lie in $B_n$. Two signed permutations are $r$-cycle equivalent (respectively, $r$-cocycle equivalent) when they are related by the corresponding cycle relations (respectively, their inverses are $r$-cycle equivalent). Conjecture B. If

$$b=ra,$$

then each Kazhdan–Lusztig right cell (respectively, left cell) consists exactly of the signed permutations that are $r$-cycle equivalent (respectively, $r$-cocycle equivalent). This conjecture proposes a direct description of Kazhdan–Lusztig cells in the integral-parameter case using cycle equivalence for domino tableaux; the source presents it as a reformulation of a conjecture of Geck and Lam.