The primitive-to-coprimitive comparison conjecture

Let $G$ be a nontrivial finite abelian group and $n\ge 1$. Let $H_{n,\mathrm{prim}}^{BM}(\mathbb X_n,\mathcal L_{n,G})$ be the common kernel of the nontrivial co-multiplication maps, and let $H_{n,\mathrm{coprim}}^{BM}(\mathbb X_n,\mathcal L_{n,G})$ be the quotient by the sum of the images of the nontrivial product maps. The primitive-to-coprimitive conjecture. The natural composition

$$H_{n,\mathrm{prim}}^{BM}(\mathbb X_n,\mathcal L_{n,G})\hookrightarrow H_n^{BM}(\mathbb X_n,\mathcal L_{n,G})\twoheadrightarrow H_{n,\mathrm{coprim}}^{BM}(\mathbb X_n,\mathcal L_{n,G})$$

is an isomorphism. This is proposed as the companion statement to the coprimitive decomposition conjecture.