Generalized Springer correspondence conjecture for graded Hecke algebras of type $B_n$

Let $R_0$ be a root system of type $B_n$, let $\bbH$ be the associated graded Hecke algebra with long-root label $k_1\neq 0$ and short-root label $k_2=mk_1$, where $m\in\frac{1}{2}\mathbb{Z}$. Let $\mathcal{P}_{n,2}$ parametrize the irreducible representations of the Weyl group $W_0$, and let $P^m_{A,B}$ be the Green functions defined using the preorder determined by $a_m$ and $\sim_m$. Write $M_A^m$ for the corresponding irreducible tempered module with real central character, and let $V_B$ denote the irreducible $W_0$-module indexed by $B$. Generalized Springer correspondence conjecture. The Green functions are independent of the chosen refinement of the preorder and are polynomials satisfying

$$P^m_{A,B}(q)=\sum_{l\geq 0}P^{m;l}_{A,B}q^l,\qquad \deg P^m_{A,B}\leq a_m(A).$$

There is a bijection

$$\widehat{\bbH}^{\mathrm{temp}}_{\mathrm{rcc}}\longleftrightarrow\widehat{W_0},$$

written $M_A^m\leftrightarrow A$, uniquely determined by the occurrence of $\chi_A\otimes\epsilon$ in $M_A^m$ when $k_1>0$ and of $\chi_A$ when $k_1<0$. The modules are graded, with degree-$l$ part

$$M_A^{m;l}\simeq\begin{cases} \displaystyle\sum_B P^{m;l}_{B,A}V_B\otimes\epsilon,&k_1>0,\\ \displaystyle\sum_B P^{m;l}_{B,A}V_B,&k_1<0, \end{cases}$$

and $M_A^m$ and $M_B^m$ have the same central character if and only if $A\sim_m B$. This conjecturally extends the equal-label Springer-module description to unequal parameters; the paper reports verification for $B_3$ and $B_4$ at special parameters, while the general case remains open.