Lusztig's multiplicity conjecture for baby Verma modules

Let $R$ be a reduced, irreducible, finite root system, $X$ its weight lattice, and $k$ a field whose characteristic is bigger than the Coxeter number of $R$. Let $\mathfrak g$ be the Lie algebra associated to $R$ over $k$, let $\widehat{\mathcal W}$ be the affine Weyl group, and let $\rho\in X$ be the half-sum of the positive roots. For $\lambda\in X$, let $\Delta(\lambda)$ be the baby Verma module and $L(\lambda)$ its simple quotient. Writing $p_{x,w}$ for the relevant Kazhdan--Lusztig polynomial, Lusztig's multiplicity conjecture. For all $x,w\in\widehat{\mathcal W}$, one has

$$[\Delta(x(\rho)-\rho):L(w(\rho)-\rho)]=p_{x,w}(1).$$

This is the multiplicity formulation of Lusztig's conjecture on the irreducible rational characters of the simply connected, connected algebraic group over $k$ with root system $R$. The paper describes it as equivalent to Lusztig's conjecture, but the supplied text gives no resolution status for the assertion itself.