The categorical p-half-twist conjecture

Fix a Weyl group $W$ and a prime $p$. Let $\mathcal{H}^p(W)$ be the Hecke category in characteristic $p$, with indecomposable objects $C_w$, and let $\operatorname{HT}_W$ be the half-twist Rouquier complex. For a $p$-cell $\lambda$, let $\mathcal{I}^p_{<\lambda}$ be the corresponding ideal. Assume the invariants $\mathbf{x}^p$, $\mathbf{c}^p$ and the involution $\operatorname{Schu}^p_L$ are defined as in the preceding categorical half-twist conjecture.

Categorical p-half-twist conjecture. For any $w$ in the $p$-cell $\lambda$,

$$\operatorname{HT}_W\otimes C_w\cong\left(\ldots_{\mathcal{I}^p_{<\lambda}}\rightarrow C_{\operatorname{Schu}^p_L(w)}(\mathbf{x}^p(\lambda))[\mathbf{c}^p(\lambda)]\rightarrow0\right).$$

Thus the characteristic-$p$ half twist should have the analogous minimal-complex behavior on $p$-cells. The paper presents this as open and emphasizes that characteristic-$p$ Rouquier complexes are largely unstudied.