Coxeter braid complex decomposition conjecture

Let $W$ be a finite Coxeter group with simple reflections $S$, let $C_\epsilon$ be the horizontal trace of the Rouquier complex associated with the Coxeter braid determined by $\epsilon\in\{\pm1\}^{|S|}$, and let $\mathrm{Cube}_W$ be the corresponding Coxeter Koszul complex. Let $p_\epsilon$ be the idempotent projecting to the Solomon summand indexed by $\epsilon$, and let $|\epsilon|_+$ denote the number of positive entries. Coxeter braid decomposition conjecture. In $K^b(\operatorname{hTr}(\operatorname{SBim}))$,

$$C_\epsilon[|\epsilon|_+]\simeq p_\epsilon\operatorname{hTr}(\mathrm{Cube}_W).$$

This is the explicit finite-Coxeter analogue of the type-$A$ annular Coxeter calculation and would describe the corresponding traced Rouquier complexes through Solomon idempotents. The supplied context does not provide a resolution, so the conjecture remains open here.