Optimal nilpotency-index bound for the Fibonacci restricted Lie algebra

Let $0\ne a\in\mathbf L$ be written in the basis as $a=\sum_{j=n}^{m}r_{j-3}v_j$, with $r_j\in R$. The preceding result gives the bound $a^{2^{m-n+2}}=0$. Optimality conjecture. This bound is optimal, as witnessed by the elements

$$a=v_n+v_{n+1}+\cdots+v_m.$$

The claim concerns the sharpness of the nilpotency-index estimate for the Fibonacci restricted Lie algebra.