The p-cell eigenmap conjecture

Fix a Weyl group $W$ and a prime $p$, and work in the characteristic-$p$ Hecke category. Let $\operatorname{FT}_W$ be the full-twist Rouquier complex, let $\mathbbm{1}$ be the monoidal unit, and let $C_w$ be the indecomposable object indexed by $w$. For each $p$-cell $\lambda$, let $\mathcal{I}^p_{<\lambda}$ be the ideal of strictly lower $p$-cells. Assume the categorical $p$-half-twist conjecture.

The p-cell eigenmap conjecture. For every $p$-cell $\lambda$, there exists a chain map

$$\alpha_\lambda:\mathbbm{1}(2\mathbf{x}(\lambda))[2\mathbf{c}(\lambda)]\longrightarrow\operatorname{FT}_W$$

which is a $\lambda$-eigenmap, meaning that for every $w\in\lambda$, the map $\alpha_\lambda\otimes C_w$ is an isomorphism modulo $\mathcal{I}^p_{<\lambda}$. This conjecture proposes eigenmaps for the additional $p$-cells arising in characteristic $p$; the paper does not provide a general proof.