The Laplacian invertibility conjecture for the flag complexes

Let $d_\Delta$ and $d_\nabla$ be the differentials of the two flag complexes associated with the co-multiplication and multiplication maps, and set

$$\bm{\Delta}:=d_{\Delta}\circ d_{\nabla}+d_{\nabla}\circ d_{\Delta}.$$

The Laplacian invertibility conjecture. On each term of the complex, the operator $\bm{\Delta}$ is invertible after tensoring with $\mathbb Q$, except on the first term $\mathcal M_n^-(G)$. This is proposed as a sufficient condition for the cohomology-vanishing conjectures for the two complexes.