Existence of limits for products of equal powers

Let $G$ be a group equipped with the bi-invariant metric norm $\\|\cdot\\|$. For elements $g_1,\dots,g_n\in G$, consider the sequence of normalized norms of products of equal powers. Existence conjecture. Given $g_1,\dots,g_n\in G$, the limit

$$\lim_{k\to\infty}\frac{1}{k}\|g_1^k\cdots g_n^k\|$$

exists. The conjecture would imply injectivity of the homomorphisms from the relevant asymptotic completion into directional asymptotic cones; the boundedness assumption on the commutator subgroup provides a sufficient condition for this existence, while the general case is left open.