Wiegold's conjecture for T_k-systems of nonabelian simple groups

Let $F_k$ be a free group of rank $k$, and let $G$ be a group. A subgroup $N\leq F_k$ with $F_k/N\cong G$ is a $G$-defining subgroup; write $\Sigma_k(G)$ for the set of all such subgroups. The connected components of the graph $\Sigma_k(G)$ under Nielsen moves are called $T_k$-systems. Wiegold's conjecture. For every $k\geq 3$ and every nonabelian simple group $G$, there exists only one $T_k$-system, i.e. the graph $\Sigma_k(G)$ is connected. This conjecture concerns the action of $\mathsf{Aut}(F_k)$ on group presentations and is known for several families of finite simple groups, including the cases listed in the paper. The paper proves it for the small Ree groups $^2\mathsf{G}_2(3^{2e+1})$ for all $e\geq 1$ and $k\geq 5$, while the general conjecture remains open.