Aluffi's Euler characteristic conjecture for graph hypersurface complements

Let $\Gamma$ be a finite graph with graph hypersurface $X_\Gamma\subset\mathbb P^{n-1}$, and write $Y_\Gamma=\mathbb P^{n-1}\smallsetminus X_\Gamma$ for its projective hypersurface complement. Aluffi's conjecture. The Euler characteristic satisfies

$$\chi(Y_\Gamma)\in\{0,1,-1\}.$$

This conjecture had been confirmed by computer calculations for many sufficiently small graphs and for some other classes of graphs. If true, it would imply that the Euler characteristics of the graph hypersurfaces $X_\Gamma$ are always non-negative.