3 problems

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 $W_n$ tha…

Conjecture A+. Let $r\geqslant 0$ and assume that $ra<b<(r+1)a$. Then the Kazhdan–Lusztig left, right, and two-sided cells for these parameters coincide with those obtained for the…

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 s…