Euler-characteristic asymptotics for the Fibonacci Lie algebra

Let $\mathcal L$ be the Fibonacci Lie algebra, let $\lambda=(1+\sqrt 5)/2$, and let $\mathbf E(\mathcal L,t)$ denote its Euler characteristic. Euler-characteristic asymptotic conjecture. There exist constants $C_1,C_2>0$ such that, as $t\to1-0$,

$$\exp\!\left(-\frac{C_1}{(1-t)^{\log_\lambda 2}}\right)\le \mathbf E(\mathcal L,t)\le \exp\!\left(-\frac{C_2}{(1-t)^{\log_\lambda 2}}\right).$$

This would quantify how the Euler characteristic approaches zero near its singularity and could have applications to the homology of $\mathcal L$.