Nonexistence of the normalized growth limit for the Fibonacci Lie algebra

Let $\mathcal L$ be the Fibonacci Lie algebra, let $\gamma_{\mathcal L}(n)$ denote its regular growth function, and let $\lambda=(1+\sqrt 5)/2$. Growth-limit conjecture. The normalized growth function has no limit:

$$\not\!\exists\lim_{n\to\infty}\frac{\gamma_{\mathcal L}(n)}{n^{\log_\lambda 2}}.$$

Earlier estimates establish the corresponding growth exponent, while the nonexistence of this limit remains a finer asymptotic question.