  Let $\mathcal S$ be a multiset of integers. We say $\mathcal S$ is a
$\textit{zero-sum sequence}$ if the sum of its elements is 0. We study zero-sum
sequences whose elements lie in the interval $[-k,k]$ such that no subsequence
of length $t$ is also zero-sum. Given these restrictions, Augspurger, Minter,
Shoukry, Sissokho, Voss show that there are arbitrarily long $t$-avoiding,
$k$-bounded zero-sum sequences unless $t$ is divisible by
$\mathrm{LCM}(2,3,4,\dots,2k-1)$. We confirm a conjecture of these authors that
for $k$ and $t$ such that this divisibility condition holds, every zero-sum
sequence of length at least $t+k^2-k$ contains a zero-sum subsequence of length
$t$, and that this is the minimal length for which this property holds.
