The pro-homology conjecture for cosheaf homology satellites

Let $X$ be a topological space, let $OPEN(X)$ be its standard site, and let $NORM(X)$ be the site of normal open sets. Write $H_n(X,A_{\#})$ for the cosheaf homology and $L_nH_0(X,A_{\#})$ for the left satellites of $H_0$.

Pro-homology conjecture. On the standard site $OPEN(X)$, if $X$ is Hausdorff paracompact, then

H_n\left(X,\operatorname{pro}\relax\protect\ifmmode\expandafter\text@\else\expandafter\text\fi{-}H_0(\bullet,A)\right)=H_n(X,A_{\#}):=L_nH_0(X,A_{\#})\simeq \operatorname{pro}\relax\protect\ifmmode\expandafter\text@\else\expandafter\text\fi{-}H_n(X,A).

The same isomorphisms should hold on $NORM(X)$ for all topological spaces. This conjecture identifies the derived cosheaf-theoretic construction with shape pro-homology under the stated hypotheses; the paper notes that general spaces may lack suitable polyhedral expansions, so the unrestricted comparison remains unavailable.