Coarse mapping-space model for coarse homotopy groups

Let $(X,\mathcal{C},\mathcal{B})$ be a bornological coarse space, let $\omega$ be a base ray, and let $n\in\mathbb{N}_0$. Define

$$c'S^n=\{(hx,h)\in\mathbb{R}^{n+1}\times[0,\infty)\mid x\in S^n,2 h\in[0,\infty)\}.$$

Coarse mapping-space conjecture. The coarse homotopy group $\pi_n^c(X,\omega)$ is isomorphic to the group, or pointed set when appropriate, of base-ray-preserving coarse homotopy classes of base-ray-preserving coarse maps $(c'S^n,c\{x_0\})\to(X,\omega)$. This would provide a direct mapping-space description of coarse homotopy groups and is used in the proposed construction of a coarse Hurewicz map.