Full-twist prediagonalization conjecture for finite Coxeter systems

Let $(W,S)$ be a finite Coxeter system, let $\operatorname{FT}(W)$ be the Rouquier complex of the full-twist braid, and let $\lambda$ range over the two-sided cells of $W$. Full-twist prediagonalization conjecture. For each two-sided cell $\lambda$ of $W$, there is a shift $\gamma_\lambda$ and a chain map $\alpha_\lambda:\gamma_\lambda\1\rightarrow\operatorname{FT}(W)$, giving $\operatorname{FT}(W)$ a prediagonalizable structure. The conjecture is known for finite dihedral groups, while the general finite-Coxeter case remains open.