Pappas–Rapoport–Smithling conjecture on maximal elements in cocharacter classes

Let $G$ be a connected reductive group over $\breve{F}$, let $T$ be a maximal torus containing a maximal $\breve{F}$-split torus of $G$, let $W=W(G,T)$ be the absolute Weyl group, and let $X_*(T)_I$ be the coinvariants of the cocharacter lattice under inertia. Fix a geometric conjugacy class of cocharacters $\{\mu\}\subset X_*(G)$, and let $\Lambda_{\{\mu\}}$ be the image in $X_*(T)_I$ of the elements of $\{\mu\}$ whose images in $X_*(T)\otimes\mathbb R$ lie in the closure of a Weyl chamber corresponding to a $\breve{F}$-rational Borel subgroup. Write $\{\bar{\mu}\}$ for the image of the $W$-conjugacy class $\{\mu\}$ in $X_*(T)_I$, and equip $X_*(T)_I$ with the Bruhat order.

Pappas–Rapoport–Smithling conjecture. The set of maximal elements in $\{\bar{\mu}\}$ with respect to the Bruhat order is precisely the set $\Lambda_{\{\mu\}}$.

This conjecture characterizes the maximal elements of the image of a cocharacter conjugacy class in the inertia coinvariants using the elements arising from rational Weyl chambers. The source presents it as a conjecture of Pappas, Rapoport, and Smithling; no resolution status is supplied here.