The satellite conjecture for pro-homotopy

Let $X$ be a space, let $NORM(X)$ be the site of normal coverings of $X$, and let $OPEN(X)$ be the standard site. Let $S$ be a set, write $S_{\#}$ for the associated constant cosheaf, and write pro\relax\protect\ifmmode\expandafter\text@\else\expandafter\text\fi{-}{\Greekmath 0119}_n(X) for pro-homotopy. The satellite conjecture for pro-homotopy. On $NORM(X)$, the non-abelian left satellites of $H_0$ are naturally isomorphic to pro-homotopy:

H_n(X,S_{\#})=H_n(X,S\times pro\relax\protect\ifmmode\expandafter\text@\else\expandafter\text\fi{-}{\Greekmath 0119}_0)\simeq S\times pro\relax\protect\ifmmode\expandafter\text@\else\expandafter\text\fi{-}{\Greekmath 0119}_n(X), H_n(X,(\mathbf{pt})_{\#})=H_n(X,pro\relax\protect\ifmmode\expandafter\text@\else\expandafter\text\fi{-}{\Greekmath 0119}_0)\simeq pro\relax\protect\ifmmode\expandafter\text@\else\expandafter\text\fi{-}{\Greekmath 0119}_n(X).

If $X$ is Hausdorff paracompact, the same isomorphisms should also exist for $OPEN(X)$. This is proposed as a connection between non-abelian cosheaf homology and shape theory; its resolution is not specified in the source.