The Hecke action conjecture for the principal block

Let $G$ be a connected reductive algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p$ strictly bigger than the Coxeter number of $G$. Let $W_{\mathrm{aff}}$ be the associated affine Weyl group, with simple reflections $S_{\mathrm{aff}}$, and let $\mathsf{Rep}(G)$ be the category of finite-dimensional algebraic $G$-modules. Write $\mathsf{Rep}_0(G)$ for its principal block, and let $\mathsf{D}_{\mathrm{BS}}$ be the Hecke category attached to $(W_{\mathrm{aff}},S_{\mathrm{aff}})$. For each $s\in S_{\mathrm{aff}}$, let $B_s$ denote the object of $\mathsf{D}_{\mathrm{BS}}$ naturally associated with $s$. Hecke action conjecture. There exists a $\Bbbk$-linear right action of the monoidal category $\mathsf{D}_{\mathrm{BS}}$ on $\mathsf{Rep}_0(G)$ such that, for every $s\in S_{\mathrm{aff}}$, the action of $B_s$ is via a functor isomorphic to the wall-crossing functor associated with $s$. This conjecture would express the combinatorics of the principal block of $\mathsf{Rep}(G)$ through the $p$-Kazhdan–Lusztig combinatorics of the Hecke category, and would imply a character formula for indecomposable tilting modules in the principal block. The source does not state whether the conjecture has been resolved.