The cuspidal, interior, and primitive cohomology conjecture

Let $G$ be a nontrivial finite abelian group and $n\ge 1$. In the Borel–Moore homology $H_n^{BM}(\mathbb X_n,\mathcal L_{n,G})$, let $H_{n,\int}(\mathbb X_n,\mathcal L_{n,G})$ be the image of ordinary homology, and let $H_{n,\mathrm{prim}}^{BM}(\mathbb X_n,\mathcal L_{n,G})$ be the common kernel of all nontrivial co-multiplication maps. The cuspidal–primitive conjecture.

$$H_{n,\mathrm{cusp}}(\mathbb X_n,\mathcal L_{n,G})=H_{n,\int}(\mathbb X_n,\mathcal L_{n,G})=H_{n,\mathrm{prim}}^{BM}(\mathbb X_n,\mathcal L_{n,G}).$$

The paper describes this as an essentially guessed identification, relating cuspidal, interior, and primitive classes.