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Let G be a finite abelian group of rank r and let X be a zero-sum free sequence over G whose support supp(X) generates G. In 2009, Pixton proved that $\left|\Sigma \left(X\right)\right|\ge 2^{r-1}\left(\right|X|-r+2)-1$ for r ≤ 3. We show that this result also holds for abelian groups G of rank 4 if the smallest prime p dividing |G| satisfies p ≥ 13.