On a classical zero-sum invariant II: Disproof of a long-standing conjecture

For a nontrivial finite abelian group $G$, let $ν(G)$ be the smallest integer $\ell$ such that every zero-sum free sequence $T$ over $G$ of length at least $\ell$ has the following property: all nonzero elements of $G$ that do not occur as a subsequence sum of $T$ lie in a proper coset of some subgr