Topological Zimmer conjecture for torsion-free finite-index subgroups on Euclidean spaces

Let $n\geq 3$, let $H$ be a torsion-free finite-index subgroup of $\mathrm{SL}_{n}(\mathbb{Z})$, and let $2\leq r<n$. Consider an action of $H$ by homeomorphisms on $\mathbb{R}^r$, equivalently a homomorphism

$$H\longrightarrow \operatorname{Homeo}(\mathbb{R}^r).$$

Topological Zimmer conjecture for torsion-free subgroups. Every such action factors through a finite group.

This is presented as an open question because the arguments proving finiteness in many compact cases use torsion elements of $\mathrm{SL}_{n}(\mathbb{Z})$, which are absent from $H$.