Exponential Nielsen growth conjecture for infinite finitely generated groups

Let $G$ be an infinite finitely generated group. The exponential Nielsen growth conjecture asserts that there are an integer $n$ and a generating $n$-tuple $S\in\Gamma_n(G)$ such that the connected component $\Gamma_n(G,S)$ has exponential growth. This formulation allows product replacement graphs to have multiple connected components and requires growth in only one component. It is weaker than the main conjecture and is known for groups of polynomial or exponential growth, while the general case remains open.