Milnor exact sequence for coarse homology of nested compacta

Let $Y$ be a compact Hausdorff space and let $(Y_i)_{i\in\mathbb{N}}$ be a decreasing family of closed subspaces, with intersection given the subspace topology. Let $H\mathcal{X}\mathcal{O}$ denote the coarse homology theory used in the source. Coarse-homology Milnor-sequence conjecture. For every $n\in\mathbb{N}_0$, there is an exact sequence

$$0\to{\lim}^{1}_{i\in\mathbb{N}}H\mathcal{X}_{n+1}\mathcal{O}(Y_i)\to H\mathcal{X}_{n}\mathcal{O}\left(\bigcap_{i\in\mathbb{N}}Y_i\right)\to\lim_{i\in\mathbb{N}}H\mathcal{X}_{n}\mathcal{O}(Y_i)\to0.$$

This is the proposed inverse-limit exact sequence for coarse homology under decreasing intersections of compact spaces.