Gao's conjecture for the zero-sum invariant

Prove or refute Gao's conjecture that \(\nu(G)=\mathsf d(G)-1\) for every nontrivial finite abelian group \(G\).

Background

The invariant $\nu(G)$ is defined as the least length threshold such that every sufficiently long zero-sum-free sequence has all its missing nonzero subsequence sums contained in a proper coset of a subgroup. The paper establishes the equality $\nu(G)=\mathsf d(G)-1$ for cyclic groups, finite abelian $p$-groups, and several additional families, including certain groups of the forms $C_2^2\oplus C_{2n}$ and $C_2^4\oplus C_{2n}$.

Despite these results, the equality remains unresolved for general nontrivial finite abelian groups. The authors emphasize that the conjecture appears difficult and may be beyond current methods, making it the paper's principal explicitly stated general open problem.