Coarse Hurewicz map

Let $(X,d)$ be a proper metric space and let $n\in\mathbb{N}_0$. Let $H\mathcal{X}_*(X)$ denote coarse homology. Coarse Hurewicz conjecture. There is a natural morphism

$$\mathcal{H}:\pi_n^c(X)\to H\mathcal{X}_{n+1}(X)$$

that makes the diagram comparing coarse homotopy with the colimit of end homotopy groups of the covers $|\mathcal{U}_h|$ commute. This would construct a coarse analogue of the ordinary Hurewicz map.