Let $\Gamma_n$ denote the isometry group of the regular tessellation of $\mathbb{R}^n$ by $n$-cubes, i.e. $\Gamma_n= \left( \bigoplus\limits_{i=1}^n \mathbb{D}_\infty \right) \rtimes S_n$. Now, let $C_n$ be an affine Coxeter group of the same dimension, say $\widetilde{A_n}$.
Does there exist a morphism $C_n \to \Gamma_n$ with infinite image?
Of course, $C_n$ contains $\mathbb{Z}^n$ as a finite-index subgroup, which clearly injects into $\Gamma_n$, but is it possible to define a similar morphism on the whole group? I verified that this is not possible for $n=2$, but the argument does not extend to higher dimenions.