The uniform homeomorphism group conjecture for Euclidean spaces

Let $\mathcal H^u(\mathbb R^n)_b$ denote the group of bounded uniform homeomorphisms of Euclidean space $\mathbb R^n$, equipped with the sup-metric, and let $\ell_\infty$ be the Banach space of bounded real sequences. Uniform homeomorphism group conjecture. The space $\mathcal H^u(\mathbb R^n)_b$ is homeomorphic to $\ell_\infty$ for any $n\geq 1$. This extends the known one-dimensional result that $\mathcal H^u(\mathbb R)_b$ is homeomorphic to $\ell_\infty$; the conjecture asks for the corresponding topological classification in every positive dimension.