On a question of Dikranjan and Zava

Let $G$ be a non-discrete countable metrizable abelian topological group endowed with the coarse structure $\mathcal{C}$ generated by compact subsets of $G$. We prove that $asdim (G, \mathcal{C} ) = \infty$. For an infinite cyclic subgroup $G$ of the circle, this answers a question of Dikranjan and Zava [3].