Non-nilpotence conjecture for the associative hull of the Fibonacci Lie algebra

Let $p=2$, let $R$ be the truncated polynomial ring from the construction, let $\mathcal L=\operatorname{Lie}(v_1,v_2)\subset\operatorname{Der}R$ be the Fibonacci Lie algebra, and let $\mathbf A=\operatorname{Alg}(v_1,v_2)$ be its associative hull. Filter $\mathbf A$ by degree in the generators, and write $\operatorname{gr}\mathbf A$ for the resulting restricted Poisson algebra. Associative-hull non-nilpotence conjecture. The restricted Poisson algebra $\operatorname{gr}\mathbf A$ is not nil; consequently, $\mathbf A$ is not nil. The proposed implication is intended to address the unresolved nilpotency question for the associative hull, in analogy with associative-algebra versions of self-similar group examples.