One-stabilization conjecture for generating graphs of finite groups

Let $G$ be a finite group, let $(\underline t)=(t_1,\ldots,t_n)$ and $(\underline s)=(s_1,\ldots,s_n)$ be elements of $\tilde X_n(G)$, the graph of generating $n$-tuples. Consider the tuples obtained by adjoining the identity element:

$$(t_1,\ldots,t_n,1),\qquad (s_1,\ldots,s_n,1).$$

One-stabilization conjecture. The tuples $(t_1,\ldots,t_n,1)$ and $(s_1,\ldots,s_n,1)$ are connected in $\bar X_{n+1}(G)$.

This is presented as an intermediate, more feasible form of Wiegold's conjecture. The statement is known for finite simple groups and finite solvable groups, but remains open for general finite groups.