The standard-homology characterization of the small Heisenberg homology

Let $\mathcal{H}_g^{V(r)}$ be the small Heisenberg homology, let $\mathcal{H}_g^{V,\dagger}$ be the direct sum of the standard twisted homology groups, and let $\iota_g^V:\mathcal{H}_g^{V,\dagger}\to\mathcal{H}_g^V$ be the natural map induced by inclusion of standard chains into Borel–Moore chains. Standard-homology characterization conjecture.

$$\mathcal{H}_g^{V(r)}=\iota_g^V\left(\mathcal{H}_g^{V,\dagger}\right).$$

The conjecture seeks an intrinsic characterization of the finite-dimensional small Heisenberg homology using standard, rather than Borel–Moore, homology; the source presents it as unresolved.