The cohomological concentration conjecture for the flag complex

Let $G$ be a finite abelian group, let $\mathcal G_\bullet$ range over flags of subgroups of $G$, and let $\mathcal M_n^-(G)$ denote the antisymmetric symbol module. The flags and the co-multiplication maps define a complex whose differential is $d_\Delta$. The flag-complex concentration conjecture. After tensoring with $\mathbb Q$, the cohomology of this complex is concentrated in degree $0$. The paper notes that this conjecture, together with the analogous vanishing for the complex with differential $d_\nabla$, would follow from invertibility of the associated Laplacian away from the first term.