Riche–Williamson conjecture on the Hecke action on the principal block

Let $\operatorname{Rep}_0$ be the principal block of finite-dimensional algebraic representations, let $\mathcal H$ be the Hecke category, and let $\Theta_s$ be the wall-crossing functor associated with each simple reflection $s$. Riche–Williamson conjecture. The category $\operatorname{Rep}_0$ is a right module category over $\mathcal H$, with the object $\mathcal E_s$ acting by the functor $\Theta_s$. This conjecture asks for a categorical lift of the affine Weyl group action on the Grothendieck group, realizing the anti-spherical module through wall-crossing functors; no resolution is given in the supplied text.