The locally finite satellite conjecture for singular homology and homotopy

Let $X$ be a pointed finite, or more generally locally finite, topological space. Let $H_n(X,G_{\#})$ denote cosheaf homology, let $L_nH_0(X,G_{\#})$ denote the left satellites of $H_0$, and write ${\Greekmath 0119}_n(X)$ for the homotopy groups.

Locally finite satellite conjecture. The left satellites are naturally isomorphic to singular homology:

$$H_n(X,G_{\#}):=L_nH_0(X,G_{\#})\simeq H_n^{\operatorname{sing}}(X,G).$$

Moreover, the nonabelian left satellites are naturally isomorphic to homotopy groups:

$$H_n(X,S_{\#})\simeq S\times{\Greekmath 0119}_n(X),\qquad H_n\left(X,(\mathbf{pt})_{\#}\right)\simeq{\Greekmath 0119}_n(X).$$

The conjecture is illustrated by a finite-circle example and contrasts with the comparatively trivial pro-homology and pro-homotopy of locally finite spaces.