Acyclicity conjecture for the extra differential on hairy graph complexes with m = -1

Let $\mathrm{HGC}_{-1,n}$ be the hairy graph complex with hair degree parameter $m=-1$ and arity parameter $n$, equipped with its standard differential $\delta$ and an extra differential $\Delta$. Acyclicity conjecture. For every $n$, the deformed complex is acyclic:

$$H\left(\mathrm{HGC}_{-1,n},\delta+\Delta\right)=0.$$

This conjecture concerns the extra differential for odd $m$, whose existence was conjectured in the preceding work; the resulting deformation is intended to make the hairy graph complex acyclic and to support the spectral-sequence method for studying its standard cohomology.