The S-arithmetic polynomial filling conjecture

Let $G$ be an $S$-arithmetic group in arbitrary characteristic, and let the filling functions and rank be understood as in the cited Lie-group theorem. The S-arithmetic polynomial filling conjecture. The analogous statement of the Leuzinger–Young theorem holds in the $S$-arithmetic case in arbitrary characteristic: the filling functions of an irreducible non-uniform lattice are polynomial below the rank. The result would provide the polynomial-cohomology input needed for below-rank comparison and Shapiro-type arguments in the $S$-arithmetic setting; the source gives no resolution status.