The lower-bound conjecture for the zero-sum-free sequence constant

Let $n>1$ be a positive integer, and let $p$ be the smallest prime divisor of $n$. For positive integers $n$ and $k$, let

$$h(n,k)=\min\{h(S)\mid |S|=n+k\},$$

where $S$ ranges over sequences in $\mathbb{Z}_n$ of length $n+k$ such that $0\notin\sum_n(S)$. Lower-bound conjecture. If $k\geq (n/p)-1$, then

$$h(n,k)\geq k+1.$$

This conjecture extends known lower bounds for $h(n,k)$ and addresses the range in which the previously expected bound has counterexamples. Its status is unresolved in the supplied source.