Decomposability conjecture beyond the stable range

Let $(G_p)_{p\geqslant 0}$ be a family of groups with multiplication, and let $\mathit{SP}_n$ denote the splitting complex associated with this family. Suppose that $|\mathit{SP}_n|$ is $(n-3)$-connected for all $n\geqslant 2$. Define

$$\mu\colon \bigoplus_{\substack{p+q=n\p,q\geqslant 1}} H_\ast(G_p)\otimes H_\ast(G_q)\longrightarrow H_\ast(G_n)$$

by $\mu(x\otimes y)=x\cdot y$, and define

$$\alpha\colon \bigoplus_{\substack{p+q+r=n\p,q,r\geqslant 1}} H_\ast(G_p)\otimes H_\ast(G_q)\otimes H_\ast(G_r)\longrightarrow \bigoplus_{\substack{p+q=n\p,q\geqslant 1}} H_\ast(G_p)\otimes H_\ast(G_q)$$

by $\alpha(x\otimes y\otimes z)=(x\cdot y)\otimes z-x\otimes(y\cdot z)$. Decomposability conjecture. The map $\mu$ is surjective in degrees $\ast\leqslant n-2$, and its kernel is the image of $\alpha$ in degrees $\ast\leqslant n-3$.

This conjecture predicts that sufficiently connected splitting complexes force homology far beyond the stable range to be generated by products, with all relations in the stated range arising from associativity. It was formulated from explicit computations for symmetric groups and braid groups, where the assertion holds; the general case remains open.