The assembly-generation conjecture below virtual cohomological dimension

Let $\Gamma_{n,s}$ denote the relevant graph automorphism group, and let $A_\phi$ be the assembly map associated to a gluing $\phi$ producing the graph $X_{n,s}$. Write $2n+s-3$ for the virtual cohomological dimension of $\Gamma_{n,s}$. Assembly-generation conjecture. If $k\not=2n+s-3$, then $H_k(\Gamma_{n,s})$ is generated by the images of the assembly maps $A_\phi$ over all possible gluings producing $X_{n,s}$. The examples in the paper verify this for $n=1$ and $n=2$; the conjecture concerns generation in every homological degree except the virtual cohomological dimension.