Bonnafé's semicontinuity conjecture for Kazhdan–Lusztig cells

Let $W$ be a Coxeter group with generating set $S$, and identify weight functions with integer points in a Euclidean space $V$ whose coordinates are their values on the conjugacy classes of generators. Let $\mathfrak{H}$ be a finite complete rational hyperplane arrangement in $V$, let $\mathcal{F}$ be an $\mathfrak{H}$-facet, and let $W_{\mathcal{F}}$ be the standard parabolic subgroup generated by the simple reflections whose weight is zero throughout $\mathcal{F}$. Denote by $\mathcal{C}_{\mathcal{L}}(L)$ and $\mathcal{C}_{\mathcal{LR}}(L)$ the partitions of $W$ into left and two-sided cells for a weight function $L$. Bonnafé's semicontinuity conjecture. There exists a finite complete rational hyperplane arrangement $\mathfrak{H}$ such that: (1) weight functions in the same facet induce the same left and two-sided cell partitions; and (2) for every facet $\mathcal{F}$, the cells on $\mathcal{F}$ are the smallest subsets of $W$ that are unions of the corresponding cells on all adjacent chambers and are stable under left translation, respectively two-sided translation, by $W_{\mathcal{F}}$. The conjecture predicts controlled variation of Kazhdan–Lusztig cells as the parameters vary, with changes governed by a finite rational arrangement and parabolic translation symmetries.