Full-twist diagonalization conjecture for Coxeter systems

Let $(W,S)$ be a Coxeter system, let $\operatorname{FT}(W)$ denote the Rouquier complex of its full-twist braid, and let $\lambda$ range over the two-sided cells of $W$. An idempotent is an object ${\mathbf P}_\lambda$ equipped with the corresponding projector structure. Full-twist diagonalization conjecture. There are idempotents ${\mathbf P}_\lambda$ for each two-sided cell such that $\{{\mathbf P}_\lambda,\alpha_\lambda\}$ is a diagonalization of $\operatorname{FT}(W)$, respecting the usual partial order on two-sided cells. The existence of these idempotents and the asserted diagonalization are presented as conjectural in the source.