The semi-infinite Goresky-MacPherson extension conjecture

Let $G$ be a reductive group, let $G_{\sO}$ be the positive-loop subgroup, let $\CF$ be an irreducible perverse sheaf on $G_{\sO}$, and let $i$ denote the locally closed embedding into the relevant arc-space compactification. Let $f_{i_{!*}\CF}$ be the would-be Goresky-MacPherson extension function, let $V$ be the representation associated with $\rho$, let $a_{\chi}$ act on an irreducible representation $V$ by multiplication by $q^{i(\chi,V)}$, and let $S$ be the Satake isomorphism.

Semi-infinite extension conjecture. If $\CF$ is the constant sheaf on $G_{\sO}$, then $f_{i_{!*}\CF}$ corresponds under the Satake isomorphism to $S(a_{\chi}([\operatorname{Sym}(V)]))$. If $(V,\rho)$ is irreducible, $n\in\mathbb{Z}$, $G_n=\{g\in G_{\sF}:v(\chi(g))=n\}$, and $\CF$ is supported on $G_0$, then

$$(f_{i_{!*}\CF})|_{G_n}\simeq q^{n\cdot i(\chi,V)}f_{\CF_{\rho,(n)}}.$$

The first assertion is known for the constant sheaf in the torus and irreducible cases, while the text presents the general formulation as conjectural.