A note on a Conjecture of Gao and Zhuang for groups of order $27$

The small Davenport constant $\mathsf{d}(G)$ of a finite group $G$ is defined to be the maximal length of a sequence over $G$ which has no non-trivial product-one subsequence. In this paper, we prove that $\mathsf{d}(G)=6$ for the non-abelian group of order $27$ and exponent $3$ and thereby establish a conjecture by Gao and Zhuang for this group.