Polynomial filling conjecture for S-arithmetic lattices

Let $k$) be a number field, let $\mathbf{G}$ be an absolutely almost simple, $k$-isotropic $k$-algebraic group, let $S$ contain all archimedean places of $k$, and let $\Gamma<G_S$ be an $S$-arithmetic lattice. Write $\operatorname{rank}_S(\mathbf{G})=\sum_{\nu\in S}\operatorname{rank}_{k_\nu}\mathbf{G}$. The filling function $\filling_{\mathbb Z,\Gamma}^i$ measures the optimal $i$-dimensional filling volume for $\Gamma$ with integral coefficients. Polynomial filling conjecture. The filling function $\filling_{\mathbb Z,\Gamma}^i$ is polynomial for every $2\le i\le \operatorname{rank}_S(\mathbf{G})-1$. This predicts polynomial higher-dimensional filling behavior for $S$-arithmetic lattices throughout the range below the $S$-rank. The supplied text gives no resolution or supporting evidence, so the conjecture is recorded as open.