The coarse filling conjecture for S-arithmetic groups

Let $K$) be a global field, meaning a global function field or a number field. Let $G$ be an absolutely almost simple $K$-isotropic $K$-group, let $S$ be a non-empty finite set of places of $K$ containing all archimedean ones, and let $n < \sum_{P \in S} \operatorname{rk}_{K_P}(G)$. A coarse $n$-manifold $\Sigma$ in a metric space is a function from the vertices of a triangulated $n$-manifold into that metric space; its boundary, scale, volume, and topological type are defined as in the setup above. Coarse filling conjecture. For every $r_1>0$, there exist a linear polynomial $f$ and $r_2>0$ such that, whenever $\Sigma \subseteq \prod_{P \in S}G(K_P)$ is a coarse $n$-manifold of scale $r_1$ with $\partial\Sigma\subseteq G(\mathcal{O}_S)$, there is a coarse $n$-manifold $\Sigma'\subseteq G(\mathcal{O}_S)$ of scale $r_2$ and identical topological type such that $\partial\Sigma'=\partial\Sigma$ and

$$\operatorname{vol}(\Sigma')\leq f(\operatorname{vol}(\Sigma)).$$

If true, this would give another proof of the rank theorem for $S$-arithmetic groups, and it concerns filling properties below the rank threshold. The source does not provide evidence resolving the conjecture.