Families and Kazhdan–Lusztig cells for unequal parameters

Let $W$ be a finite Coxeter group with weight function $L$. Lusztig's families partition the irreducible representations $\text{Irr}(W)$, and the Kazhdan–Lusztig preorder $\leq_{\mathcal{LR}}$ induces another partition of $\text{Irr}(W)$. Lusztig's conjecture. These two partitions agree. This is known for equal parameters and has been verified by explicit computation in types $I_2(m)$ and $F_4$; the general unequal-parameter case remains open.