Bruhat-order characterization for affine Weyl groups of type A

Let $(\lambda_1,w_1)$ and $(\lambda_2,w_2)$ be elements of the affine Weyl group written in semidirect-product form, where $\lambda_1$ and $\lambda_2$ lie in the root lattice and $w_1$ and $w_2$ lie in the finite Weyl group. For each weight $\lambda$, let $\lambda^+$ denote the unique dominant weight in its finite Weyl-group orbit, and set $\varpi_0=0$. Bruhat-order characterization. The relation $(\lambda_1,w_1)\leq(\lambda_2,w_2)$ in Bruhat order holds if and only if

$$(\lambda_1-\varpi_i+w_1(\varpi_j))^+\leq(\lambda_2-\varpi_i+w_2(\varpi_j))^+$$

in the dominance order for all $0\leq i,j\leq n$. This gives a proposed criterion for comparing affine Weyl-group elements using only dominant representatives of the indicated weights; the supplied text does not state whether the criterion has been proved or remains open.