Nielsen equivalence in small cancellation groups

Let $G$ be a group given by the presentation [<a_1,…,a_k,b_1,… b_k,| a_i=u_i(\bar b), b_i=v_i(\bar a) \hbox{for} 1\le i\le k>,] where $k\ge 2$ and where the $u_i\in F(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_k)$ and $w_i\in F(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_k)$ are random words. Generically such a group is a small cancellation group and it is clear that $(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_k)$ and $(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_k)$ are generating $n$-tuples for $G$. We prove that for generic choices of $u_1,\mathrm{.}\mathrm{.}\mathrm{.},u_k$ and $v_1,\mathrm{.}\mathrm{.}\mathrm{.},v_k$ the “once-stabilized” tuples $(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_k,1)$ and $(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_k,1)$ are not Nielsen equivalent in $G$. This provides a counter-example for a Wiegold-type conjecture in the setting of word-hyperbolic groups. We conjecture that in the above construction at least $k$ stabilizations are needed to make the tuples $(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_k)$ and $(b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_k)$ Nielsen equivalent.