Vanishing conjecture for assembled classes under rank stabilization

Let $\phi$ be a gluing and let $A_\phi$ be its assembly map for the homology groups of the graph automorphism groups $\Gamma_{n,s}$. A class is positive-dimensional when it has homological degree greater than zero. Stabilization-vanishing conjecture. If $\phi$ is any gluing, then all positive-dimensional classes in the image of $A_\phi$ vanish after a single stabilization with respect to $n$. The paper proves this when the stabilization uses a rank-$1$ factor, but the general assertion remains open.