Lusztig's character formula conjecture

Assume that $p$ is the characteristic, $h$ is the Coxeter number, $W_{\mathrm{aff}}^\varnothing$ is the specified subset of the affine Weyl group, $\rho$ is the half-sum of positive roots, $\bullet$ denotes the affine dot action, $\Phi_+^\vee$ is the set of positive coroots, $c_{y,w}^\varnothing$ are the character coefficients, $\ell$ is the length function, $w_\circ$ is the longest element, and $h_{x,z}(1)$ denotes the relevant Kazhdan–Lusztig polynomial evaluated at $1$.

Lusztig's character formula conjecture. If $p \geq h$ and $w \in W_{\mathrm{aff}}^\varnothing$ satisfies

$$\langle w \bullet 0+\rho,\alpha^\vee\rangle\leq p(p-h+2)$$

for every $\alpha^\vee\in\Phi_+^\vee$, then

$$c_{y,w}^\varnothing=(-1)^{\ell(w)+\ell(y)}h_{w_\circ y,w_\circ w}(1)$$

for every $y\in W_{\mathrm{aff}}^\varnothing$.

The conjecture is presented as the character-formula problem for simple modules. Its status is not established by the supplied text.