The categorical half-twist conjecture

Let $W$ be a finite Coxeter group, let $\mathcal{H}^0(W)$ be its characteristic-zero Hecke category, and let $K^b(\mathcal{H}^0)$ be the bounded homotopy category. For each cell $\lambda$, let $\mathcal{I}_{<\lambda}$ be the ideal generated by strictly lower cells, and let $B_w$ be the indecomposable Soergel bimodule indexed by $w$. Let $\operatorname{HT}_W$ denote the half-twist Rouquier complex.

Categorical half-twist conjecture. If $w$ lies in the cell $\lambda$, then

$$\operatorname{HT}_W\otimes B_w\cong \left(\ldots_{\mathcal{I}_{<\lambda}}\rightarrow B_{\operatorname{Schu}_L(w)}(\mathbf{x}(\lambda))[\mathbf{c}(\lambda)]\rightarrow 0\right).$$

Equivalently, the minimal complex has one copy of $B_{\operatorname{Schu}_L(w)}$ in homological degree $\mathbf{c}(\lambda)$ and grading shift $\mathbf{x}(\lambda)$, with all other terms in strictly lower cells and strictly lower homological degrees. This conjecture was proven in type $A$; the paper notes that a proof for dihedral types was in preparation.