Weak-order maximality conjecture for affine Weyl group cores

Let $\widetilde{W}$ be a simply-laced affine Weyl group with Coxeter number $h$, let $b$ be coprime to $h$, and let $\mathcal{S}_\Phi(b)$ be the relevant Sommers region. Let $\widetilde{w}_b$ be the distinguished affine Weyl group element associated with $b$, and regard dominant elements as elements of $\widetilde{W}/W$. Weak-order maximality conjecture. The element $\widetilde{w}_b$ is maximal in the weak order on $\widetilde{W}/W$ among all dominant elements

$$\left\{\widetilde{w} \in \widetilde{W}/W: \widetilde{w}^{-1}(0) \in \mathcal{S}_\Phi(b)\right\}.$$

The conjecture strengthens the proved uniqueness of the maximum of the size statistic by predicting containment of the relevant inversion sets. It generalizes J. Vandehey's result for ordinary $(a,b)$-cores.