The rank-one quasi-isometry classification conjecture for function-field arithmetic groups

Let $K$ be a global function field, let $S$ be a nonempty set of valuations of $K$, and let $\mathbf{G}$ be a connected, absolutely simple, $K$-isotropic algebraic $K$-group of adjoint type. For each $v\in S$, write $K_v$ for the completion of $K$ at $v$, set $G_S=\prod_{v\in S}\mathbf{G}(K_v)$, and let $\mathbf{G}(\mathcal{O}_S)$ be embedded diagonally as a discrete subgroup of $G_S$. The group $\mathcal{QI}(\mathbf{G}(\mathcal{O}_S))$ denotes its quasi-isometry group, and $\operatorname{Comm}_{\operatorname{Aut}(G_S)}(\mathbf{G}(\mathcal{O}_S))$ denotes the group of topological automorphisms of $G_S$ that commensurate $\mathbf{G}(\mathcal{O}_S)$. The rank-one quasi-isometry classification conjecture. If

$$\operatorname{rank}_{K_v}(\mathbf{G})=1\quad\text{for all }v\in S,\qquad |S|>1,$$

then there is an isomorphism

$$\mathcal{QI}(\mathbf{G}(\mathcal{O}_S))\cong \operatorname{Comm}_{\operatorname{Aut}(G_S)}(\mathbf{G}(\mathcal{O}_S)).$$

The paper says that, after the higher-rank and mixed-rank cases, this reduces the remaining problem to lattice actions on products of trees. Thus this rank-one case remains open in the source.