  We show that for every $n\ge 2$ there exists a torsion-free one-ended
word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples
$(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples
$$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})\hbox{ and
}(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})$$ are not
Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic
construction.
