The unequal-parameter cell classification conjecture for type $B_n$

Let $W$ be of type $B_n$, with generators $t,s_1,\ldots,s_{n-1}$, where $t$ is not conjugate to the $s_i$, and let a positive weight function be specified by

$$b:=L(t)>0,$$ $$a:=L(s_1)=L(s_2)=\cdots=L(s_{n-1})>0.$$

Write the weight function as $(b,a,a,\ldots,a)$. The unequal-parameter cell classification conjecture. The equivalence classes of weight functions are

$$\begin{aligned} {\mathcal L}_0 &= \{ (b,a,a,\ldots,a) \mid a>b>0\},\\ {\mathcal L}_1 &= \{ (a,a,a,\ldots,a) \mid a>0\},\\ {\mathcal L}_i &= \{ (ia,a,a,\ldots,a) \mid a>0\} \qquad (2\leqslant i\leqslant n-1),\\ {\mathcal L}_{i,i-1} &= \{(b,a,a,\ldots,a) \mid ia>b>(i-1)a>0\} \qquad (2\leqslant i\leqslant n-1),\\ {\mathcal L}_{\mathrm{asymp}} &= \{ (b,a,a,\ldots,a) \mid b>(n-1)a>0\}. \end{aligned}$$

The class ${\mathcal L}_1$ is the equal-parameter case, and ${\mathcal L}_{\mathrm{asymp}}$ is the asymptotic case. This conjecture proposes a complete classification of equivalence classes of positive weight functions for type $B_n$; the asymptotic range was already treated by Bonnaf'e and Iancu, while the intermediate cases remain to be established.