Cofibre-colimit model for coarse homology at infinity

Let $E:\mathbf{BornCoarse}\to\mathbf{C}$ be a coarse homology theory, let $X$ be a uniform bornological coarse space, and let $i_\alpha:M_\alpha\to M$ be the canonical inclusions associated to a uniform shape expansion, with cofibre $C(i_\alpha)$. Assume that $\mathbf{C}$ is complete and that $E$ is additive. Cofibre-colimit conjecture. There is an equivalence

$$\operatorname{colim}_{\alpha}\left(E(\ast)\wedge\Sigma_{+}^{\infty}\right)^{lf}(C(i_\alpha))\simeq E\mathcal{O}^{\infty}(X).$$

This would identify the coarse homology at infinity of $X$ with a colimit of locally finite cofibres arising from its uniform shape model.