  Let $G=(\mathbb Z/n\mathbb Z) \oplus (\mathbb Z/n\mathbb Z)$. Let $\mathsf
{s}_{\leq k}(G)$ be the smallest integer $\ell$ such that every sequence of
$\ell$ terms from $G$, with repetition allowed, has a nonempty zero-sum
subsequence with length at most $k$. It is known that $\mathsf {s}_{\leq
2n-1-k}(G)=2n-1+k$ for $k\in [0,n-1]$, with the structure of extremal sequences
showing this bound tight determined when $k\in \{0,1,n-1\}$, and for various
special cases when $k\in [2,n-2]$. For the remaining values $k\in [2,n-2]$, the
characterization of extremal sequences of length $2n-2+k$ avoiding a nonempty
zero-sum of length at most $2n-1-k$ remained open in general, with it
conjectured that they must all have the form $e_1^{[n-1]} \boldsymbol{\cdot}
e_2^{[n-1]} \boldsymbol{\cdot} (e_1 +e_2)^{[k]}$ for some basis $(e_1,e_2)$ for
$G$. Here $x^{[n]}$ denotes a sequence consisting of the term $x$ repeated $n$
times. In this paper, we establish this conjecture for all $k\in [2,n-2]$ when
$n$ is prime, which in view of other recent work, implies the conjectured
structure for all rank two abelian groups.
