Equality of asymptotic norms in the free-group completion

Let $G$ be a group with norm $\|\cdot\|$, and let $F(G)$ denote the free group on the underlying set of $G$, with generators written $\overline g$. For $g_1,\dots,g_n\in G$, compare the normalized norm of their product in $G$ with the normalized free-group norm of the product of the corresponding powered generators. Asymptotic norm conjecture. For any $g_1,\dots,g_n\in G$,

$$\lim_{k\to\infty}\frac{1}{k}\|g_1^k\cdots g_n^k\|=\lim_{k\to\infty}\frac{1}{k}\|\overline{g_1^k}\cdots\overline{g_n^k}\|_{F(G)}.$$

This equality is presented as the additional condition needed to establish that the relevant homomorphisms are isometric embeddings; the source does not state a resolution.